Elementary Linear Algebra Anton & Rorres, 9 th Edition Lecture Set – 08 Chapter 8: Linear Transformations.

Slides:



Advertisements
Similar presentations
10.4 Complex Vector Spaces.
Advertisements

5.4 Basis And Dimension.
8.4 Matrices of General Linear Transformations
5.1 Real Vector Spaces.
Chapter 4 Euclidean Vector Spaces
6.4 Best Approximation; Least Squares
8.3 Inverse Linear Transformations
8.2 Kernel And Range.
Chapter 3: Linear transformations
Chapter 6 Eigenvalues and Eigenvectors
Linear Algebra Canonical Forms 資料來源: Friedberg, Insel, and Spence, “Linear Algebra”, 2nd ed., Prentice-Hall. (Chapter 7) 大葉大學 資訊工程系 黃鈴玲.
Elementary Linear Algebra Anton & Rorres, 9th Edition
Symmetric Matrices and Quadratic Forms
Chapter 5 Orthogonality
Orthogonality and Least Squares
Boot Camp in Linear Algebra Joel Barajas Karla L Caballero University of California Silicon Valley Center October 8th, 2008.
Chapter 6 Linear Transformations
Stats & Linear Models.
Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka Virginia de Sa (UCSD) Cogsci 108F Linear.
1 MAC 2103 Module 10 lnner Product Spaces I. 2 Rev.F09 Learning Objectives Upon completing this module, you should be able to: 1. Define and find the.
Linear Algebra Review 1 CS479/679 Pattern Recognition Dr. George Bebis.
1 February 24 Matrices 3.2 Matrices; Row reduction Standard form of a set of linear equations: Chapter 3 Linear Algebra Matrix of coefficients: Augmented.
6.1 Introduction to Linear Transformations
Chapter 4 Chapter Content
Elementary Linear Algebra Anton & Rorres, 9th Edition
4 4.4 © 2012 Pearson Education, Inc. Vector Spaces COORDINATE SYSTEMS.
Vectors in R n a sequence of n real number An ordered n-tuple: the set of all ordered n-tuple  n-space: R n Notes: (1) An n-tuple can be viewed.
Section 4.1 Vectors in ℝ n. ℝ n Vectors Vector addition Scalar multiplication.
Chapter 3 Vector Spaces. The operations of addition and scalar multiplication are used in many contexts in mathematics. Regardless of the context, however,
Chapter Content Real Vector Spaces Subspaces Linear Independence
Matrices CHAPTER 8.1 ~ 8.8. Ch _2 Contents  8.1 Matrix Algebra 8.1 Matrix Algebra  8.2 Systems of Linear Algebra Equations 8.2 Systems of Linear.
Chap. 6 Linear Transformations
Chapter 5 Eigenvalues and Eigenvectors 大葉大學 資訊工程系 黃鈴玲 Linear Algebra.
Elementary Linear Algebra Anton & Rorres, 9th Edition
1 MAC 2103 Module 7 Euclidean Vector Spaces II. 2 Rev.F09 Learning Objectives Upon completing this module, you should be able to: 1. Determine if a linear.
Introductions to Linear Transformations Function T that maps a vector space V into a vector space W: V: the domain of T W: the codomain of T Chapter.
1 Chapter 3 – Subspaces of R n and Their Dimension Outline 3.1 Image and Kernel of a Linear Transformation 3.2 Subspaces of R n ; Bases and Linear Independence.
Chapter 4 Linear Transformations 4.1 Introduction to Linear Transformations 4.2 The Kernel and Range of a Linear Transformation 4.3 Matrices for Linear.
Elementary Linear Algebra Anton & Rorres, 9 th Edition Lecture Set – 02 Chapter 2: Determinants.
5.5 Row Space, Column Space, and Nullspace
Eigenvectors and Linear Transformations Recall the definition of similar matrices: Let A and C be n  n matrices. We say that A is similar to C in case.
Chapter 4 – Linear Spaces
8.5 Similarity. SIMILARITY The matrix of a linear operator T:V V depends on the basis selected for V that makes the matrix for T as simple as possible.
4 © 2012 Pearson Education, Inc. Vector Spaces 4.4 COORDINATE SYSTEMS.
Chap. 4 Vector Spaces 4.1 Vectors in Rn 4.2 Vector Spaces
Elementary Linear Algebra Anton & Rorres, 9 th Edition Lecture Set – 07 Chapter 7: Eigenvalues, Eigenvectors.
Elementary Linear Algebra Anton & Rorres, 9th Edition
Advanced Computer Graphics Spring 2014 K. H. Ko School of Mechatronics Gwangju Institute of Science and Technology.
FUNCTIONS COSC-1321 Discrete Structures 1. Function. Definition Let X and Y be sets. A function f from X to Y is a relation from X to Y with the property.
A function is a rule f that associates with each element in a set A one and only one element in a set B. If f associates the element b with the element.
1 Chapter 8 – Symmetric Matrices and Quadratic Forms Outline 8.1 Symmetric Matrices 8.2Quardratic Forms 8.3Singular ValuesSymmetric MatricesQuardratic.
Chapter 4 Chapter Content 1. Real Vector Spaces 2. Subspaces 3. Linear Independence 4. Basis 5.Dimension 6. Row Space, Column Space, and Nullspace 8.Rank.
Chapter 6- LINEAR MAPPINGS LECTURE 8 Prof. Dr. Zafer ASLAN.
Section 4.3 Properties of Linear Transformations from R n to R m.
Chapter 5 Chapter Content 1. Real Vector Spaces 2. Subspaces 3. Linear Independence 4. Basis and Dimension 5. Row Space, Column Space, and Nullspace 6.
Boot Camp in Linear Algebra TIM 209 Prof. Ram Akella.
Chapter 4 Linear Transformations 4.1 Introduction to Linear Transformations 4.2 The Kernel and Range of a Linear Transformation 4.3 Matrices for Linear.
8.1 General Linear Transformation
8.2 Kernel And Range.
Elementary Linear Algebra Anton & Rorres, 9th Edition
8.3 Inverse Linear Transformations
Euclidean Inner Product on Rn
Linear Transformations
Elementary Linear Algebra
Elementary Linear Algebra Anton & Rorres, 9th Edition
Elementary Linear Algebra Anton & Rorres, 9th Edition
Elementary Linear Algebra Anton & Rorres, 9th Edition
Chapter 4 Linear Transformations
Eigenvalues and Eigenvectors
Elementary Linear Algebra Anton & Rorres, 9th Edition
Presentation transcript:

Elementary Linear Algebra Anton & Rorres, 9 th Edition Lecture Set – 08 Chapter 8: Linear Transformations

2015/7/2 Elementary Linear Algebra 2 Chapter Content General Linear Transformations Kernel and Range Inverse Linear Transformations Matrices of General Linear Transformations Similarity Isomorphism

2015/7/2 Elementary Linear Algebra 3 Linear Transformation Definition  If T : V  W is a function from a vector space V into a vector space W, then T is called a linear transformation from V to W if for all vectors u and v in V and all scalars c T (u + v) = T (u) + T (v) T (cu) = cT (u) In the special case where V = W, the linear transformation T : V  V is called a linear operator on V.

Linear Transformation Example (Zero Transformation)  The mapping T : V  W such that T(v) = 0 for every v in V is a linear transformation called the zero transformation. Example (Identity Operator)  The mapping I : V  I defined by I (v) = v is called the identity operator on V. 2015/7/2 Elementary Linear Algebra 4

2015/7/2 Elementary Linear Algebra 5 Orthogonal Projections Suppose that W is a finite-dimensional subspace of an inner product space V ; then the orthogonal projection of V onto W is the transformation defined by T (v) = proj W v If S = {w 1, w 2, …, w r } is any orthogonal basis for W, then T (v) is given by the formula T (v) = proj W v =  v, w 1  w 1 +  v, w 2  w 2 + ··· +  v, w r  w r This projection a linear transformation:  T(u + v) = T(u) + T(v)  T(cu) = cT(u)

2015/7/2 Elementary Linear Algebra 6 A Linear Transformation from a Space V to R n Let S = {w 1, w 2, …, w n } be a basis for an n-dimensional vector space V, and let (v) s = (k 1,, k 2,, …, k n ) be the coordinate vector relative to S of a vector v in V; thus v = k 1 w 1 + k 2 w 2 + …+ k n w n Define T : V  R n to be the function that maps v into its coordinate vector relative to S; that is, T (v) = (v) s = (k 1,, k 2,, …, k n ) Then the function T is a linear transformation:  Let u = c 1 w 1 + c 2 w 2 + …+ c n w n and v = d 1 w 1 + d 2 w 2 + …+ d n w n  Check if (u + v) s = (u) s + (v) s and (ku) s = k(u) s

2015/7/2 Elementary Linear Algebra 7 A Linear Transformation from P n to P n+1 Let p = p(x) = c 0 + c 1 x + ··· + c n x n be a polynomial in P n, and define the function T : P n  P n+1 by T (p) = T (p(x)) = xp(x) = c 0 x + c 1 x 2 + ··· + c n x n+1 The function T is a linear transformation:  For any scalar k and any polynomials p 1 and p 2 in P n we have T (p 1 + p 2 ) = T (p 1 (x) + p 2 (x)) = x (p 1 (x) + p 2 (x)) = x p 1 (x) + x p 2 (x) = T (p 1 ) + T (p 2 ) T (k p) = T (k p(x)) = x (k p(x)) = k (x p(x))= k T(p)

2015/7/2 Elementary Linear Algebra 8 A Linear Transformation Using an Inner Product Let V be an inner product space and let v 0 be any fixed vector in V. Let T : V  R be the transformation that maps a vector v into its inner product with v 0 ; that is, T (v) =  v, v 0  From the properties of an inner product  T (u + v) =  u + v, v 0  =  u, v 0  +  v, v 0   T (k u) =  k u, v 0  = k  u, v 0  = kT (u) Thus, T is a linear transformation.

2015/7/2 Elementary Linear Algebra 9 Example Let T:M nn →R be the transformation that maps an n × n matrix into its determinant; that is, T (A) = det (A) If n>1, then this transformation does not satisfy either of the properties required of a linear transformation. For example, we saw Example 1 of Section 2.3 that det (A 1 +A 2 ) ≠ det (A 1 ) + det (A 2 ) in general. Moreover, det (cA) =C n det (A), so det (cA) ≠ c det (A) in general. Thus, T is not linear transformation.

2015/7/2 Elementary Linear Algebra 10 Properties of Linear Transformation If T : V  W is a linear transformation, then for any vectors v 1 and v 2 in V and any scalars c 1 and c 2, we have T (c 1 v 1 + c 2 v 2 ) = T (c 1 v 1 ) + T (c 2 v 2 ) = c 1 T (v 1 ) + c 2 T (v 2 ) More generally, if v 1, v 2, …, v n are vectors in V and c 1, c 2, …, c n are scalars, then T (c 1 v 1 + c 2 v 2 +…+ c n v n ) = c 1 T (v 1 ) + c 2 T (v 2 ) +…+ c n T (v n ) The above equation is sometimes described by saying that linear transformations preserve linear combinations.

2015/7/2 Elementary Linear Algebra 11 Theorem Theorem 8.1  If T : V  W is a linear transformation, then T(0) = 0 T(-v) = -T(v) for all v in V T(v – w) = T(v) – T(w) for all v and w in V

2015/7/2 Elementary Linear Algebra 12 Finding Linear Transformations from Images of Basis If T : V  W is a linear transformation, and if {v 1, v 2, …, v n } is any basis for V, then the image T (v) of any vector v in V can be calculated from the images T (v 1 ), T (v 2 ), …, T (v n ) of the basis vectors. This can be done by first expressing v as a linear combination of the basis vectors, say v = c 1 v 1 + c 2 v 2 + …+ c n v n and then the transformation becomes T (v) = c 1 T (v 1 ) + c 2 T (v 2 ) + … + c n T (v n ) A linear transformation is completely determined by its images of any basis vectors.

2015/7/2 Elementary Linear Algebra 13 Example Consider the basis S = {v 1, v 2, v 3 } for R 3, where v 1 = (1,1,1), v 2 = (1,1,0), and v 3 = (1,0,0). Let T: R 3  R 2 be the linear transformation such that T (v 1 ) = (1,0), T (v 2 ) = (2,-1), T (v 3 ) = (4,3). Find a formula for T (x 1, x 2, x 3 ); then use this formula to compute T (2, -3, 5).

2015/7/2 Elementary Linear Algebra 14 Composition of T 2 with T 1 Definition  If T 1 : U  V and T 2 : V  W are linear transformations, the composition of T 2 with T 1, denoted by T 2  T 1 (read “T 2 circle T 1 ”), is the function defined by the formula (T 2  T 1 )(u) = T 2 (T 1 (u)) where u is a vector in U. Theorem  If T 1 : U  V and T 2 : V  W are linear transformations, then (T 2  T 1 ) : U  W is also a linear transformation.

2015/7/2 Elementary Linear Algebra 15 Remark The compositions can be defined for more than two linear transformations. For example, if T 1 : U  V and T 2 : V  W,and T 3 : W  Y are linear transformations, then the composition T 3  T 2  T 1 is defined by (T 3  T 2  T 1 )(u) = T 3 (T 2 (T 1 (u)))

2015/7/2 Elementary Linear Algebra 16 Chapter Content General Linear Transformations Kernel and Range Inverse Linear Transformations Matrices of General Linear Transformations Similarity Isomorphism

2015/7/2 Elementary Linear Algebra 17 Kernel and Range Recall:  If A is an m  n matrix, then the nullspace of A consists of all vector x in R n such that Ax = 0.  The column space of A consists of all vectors b in R m for which there is at least one vector x in R n such that Ax = b.  The nullspace of A consists of all vectors in R n that multiplication by A maps into 0. (in terms of matrix transformation)  The column space of A consists of all vectors in R m that are images of at least one vector in R n under multiplication by A. (in terms of matrix transformation)

Kernel and Range Definition  If T : V  W is a linear transformation, then the set of vectors in V that T maps into 0 is called the kernel of T; it is denoted by ker(T).  The set of all vectors in W that are images under T of at least one vector in V is called the range of T; it is denoted by R(T). 2015/7/2 Elementary Linear Algebra 18

2015/7/2 Elementary Linear Algebra 19 Examples If T A : R n  R m is multiplication by the m  n matrix A, then the kernel of T A is the nullspace of A and the range of T A is the column space of A. Let T : V  W be the zero transformation. Since T maps every vector in V into 0, it follows that ker(T) = V. Moreover, since 0 is the only image under T of vectors in V, we have R(T) = {0}. Let I : V  V be the identity operator. Since I (v) = v for all vectors in V, every vector in V is the image of some vector; thus, R(I) = V. Since the only vector that I maps into 0 is 0, it follows ker(I) = {0}.

2015/7/2 Elementary Linear Algebra 20 Example Let T : R 3  R 3 be the orthogonal projection on the xy-plane. The kernel of T is the set of points that T maps into 0 = (0,0,0); these are the points on the z-axis. Since T maps every points in R 3 into the xy-plane, the range of T must be some subset of this plane. But every point (x 0,y 0,0) in the xy-plane is the image under T of some point. Thus R(T) is the entire xy-plane.

2015/7/2 Elementary Linear Algebra 21 Example Let T : R 2  R 2 be the linear operator that rotates each vector in the xy-plane through the angle . Since every vector in the xy-plane can be obtained by rotating through some vector through angle , we have R(T) = R 2. The only vector that rotates into 0 is 0, so ker(T) = {0}.

2015/7/2 Elementary Linear Algebra 22 Properties of Kernel and Range Theorem  If T : V  W is linear transformation, then: The kernel of T is a subspace of V. The range of T is a subspace of W.

Properties of Kernel and Range Definition  If T : V  W is a linear transformation, then the dimension of the range of T is called the rank of T and is denoted by rank(T).  The dimension of the kernel is called the nullity of T and is denoted by nullity(T). Theorem  If A is an m  n matrix and T A : R n  R m is multiplication by A, then: nullity (T A ) = nullity (A) rank (T A ) = rank (A) 2015/7/2 Elementary Linear Algebra 23

2015/7/2 Elementary Linear Algebra 24 Example Let T A : R 6  R 4 be multiplication by Find the rank and nullity of T A In Example 1 of Section 5.6 we showed that rank (A) = 2 and nullity (A) = 4. (use reduced row-echelon form, etc.) Thus, from Theorem 8.2.2, rank (T A ) = 2 and nullity (T A ) = 4.

2015/7/2 Elementary Linear Algebra 25 Example Let T : R 3  R 3 be the orthogonal projection on the xy- plane. From Example 4, the kernel of T is the z-axis, which is one-dimensional; and the range of T is the xy-plane, which is two-dimensional. Thus, nullity (T) = 1 and rank (T) = 2.

2015/7/2 Elementary Linear Algebra 26 Dimension Theorem for Linear Transformations Theorem  If T : V  W is a linear transformation from an n- dimensional vector space V to a vector space W, then rank(T) + nullity(T) = n Remark  In words, this theorem states that for linear transformations the rank plus the nullity is equal to the dimension of the domain.

2015/7/2 Elementary Linear Algebra 27 Chapter Content General Linear Transformations Kernel and Range Inverse Linear Transformations Matrices of General Linear Transformations Similarity Isomorphism

2015/7/2 Elementary Linear Algebra 28 One-to-One Linear Transformation A linear transformation T : V  W is said to be one- to-one if T maps distinct vectors in V into distinct vectors in W. Examples  If A is an n  n matrix and T A : R n  R n is multiplication by A, then T A is one-to-one if and only if A is an invertible matrix (Theorem 4.3.1).

Example Let T : R 2  R 2 be the linear operator that rotates each vector in the xy-plane through an angle . We showed that ker(T) = {0} and R(T) = R 2.  Thus, rank(T) + nullity(T) = = /7/2 Elementary Linear Algebra 29

2015/7/2 Elementary Linear Algebra 30 Theorem (Equivalent Statements) If T : V  W is a linear transformation, then the following are equivalent.  T is one-to-one  The kernel of T contains only zero vector; that is, ker(T) = {0}  Nullity(T) = 0

Theorem If V is a finite-dimensional vector space and T : V  V is a linear operator, then the following are equivalent.  T is one-to-one  ker(T) = {0}  Nullity(T) = 0  The range of T is V; that is, R(T) = V 2015/7/2 Elementary Linear Algebra 31

2015/7/2 Elementary Linear Algebra 32 Example Let T A : R 4  R 4 be multiplication by Determine whether T A is one to one. Solution:  det(A) = 0, since the first two rows of A are proportional  A is not invertible  T A is not one-to-one.

2015/7/2 Elementary Linear Algebra 33 Inverse Linear Transformations If T : V  W is a linear transformation, then the range of T denoted by R (T), is the subspace of W consisting of all images under T of vectors in V. If T is one-to-one, then each vector w in R(T) is the image of a unique vector v in V. This uniqueness allows us to define a new function, call the inverse of T, denoted by T – 1, which maps w back into v. The mapping T – 1 : R (T)  V is a linear transformation. Moreover, T – 1 (T (v)) = T – 1 (w) = v T – 1 (T (w)) = T – 1 (v) = w

Inverse Linear Transformations If T : V  W is a one-to-one linear transformation, then the domain of T – 1 is the range of T. The range may or may not be all of W (one-to-one but not onto). For the special case that T : V  V, then the linear transformation is one-to-one and onto. 2015/7/2 Elementary Linear Algebra 34

2015/7/2 Elementary Linear Algebra 35 Example (An Inverse Transformation) Let T : R 3  R 3 be the linear operator defined by the formula T (x 1, x 2, x 3 ) = (3x 1 + x 2, -2x 1 – 4x 2 + 3x 3, 5x x 2 – 2x 3 ). Determine whether T is one-to-one; if so, find T -1 (x 1,x 2,x 3 ). Solution:

2015/7/2 Elementary Linear Algebra 36 Theorem If T 1 : U  V and T 2 : V  W are one to one linear transformation then:  T 2  T 1 is one to one  (T 2  T 1 ) -1 = T 1 -1  T 2 -1

2015/7/2 Elementary Linear Algebra 37 Chapter Content General Linear Transformations Kernel and Range Inverse Linear Transformations Matrices of General Linear Transformations Similarity Isomorphism

2015/7/2 Elementary Linear Algebra 38 Matrices of General Linear Transformations Remark:  If V and W are finite-dimensional vector spaces (not necessarily R n and R m ), then any transformation T : V  W can be regarded as a matrix transformation.  The basic idea is to work with coordinate matrices of the vectors rather than with the vectors themselves.

2015/7/2 Elementary Linear Algebra 39 Matrices of Linear Transformations Suppose V and W are n and m dimensional vector space and B and B are bases for V and W, then for x in V, the coordinate matrix [x] B will be a vector in R n, and coordinate matrix [T(x)] B will be a vector in R m. A vector in V (n-dimensional) A vector in R n A vector in W (m-dimensional) A vector in R m x T (x) [x]B[x]B [T (x)] B T ?

2015/7/2 Elementary Linear Algebra 40 Matrices of Linear Transformations If we let A be the standard matrix for this transformation, then A [x] B = [T (x)] B The matrix A is called the matrix for T with respect to the bases B and B x T (x) [x]B[x]B [T (x)] B T T maps V into W A Multiplication by A maps R n to R m

2015/7/2 Elementary Linear Algebra 41 Matrices of Linear Transformations Let B = {u 1, …, u n } be a basis for the n-dimensional space V and B = {u 1, …, u m } be a basis for the m-dimensional space W. Consider an m  n matrix such that A [x] B = [T(x)] B holds for all vectors x in V. That is, A [x] B = [T(x)] B has to hold for the basis vectors u 1, …, u n. Thus, we need A [u 1 ] B = [T(u 1 )] B, A [u 2 ] B = [T(u 2 )] B, …, A [u n ] B = [T(u n )] B Since [u 1 ] B = e 1, [u 2 ] B = e 2, …, [u n ] B = e n

2015/7/2 Elementary Linear Algebra 42 Matrices of Linear Transformations We have Thus,, which is the matrix for T w.r.t. the bases B and B, and denoted by the symbol [T] B,B That is, and Basis for the image spaceBasis for the domain

2015/7/2 Elementary Linear Algebra 43 Matrices for Linear Operators In the special case where V = W, the resulting matrix is called the matrix for T with respect to the basis B and denoted by [T] B rather than [T] B,B. If B = {u 1, …, u n }, then we have and That is, the matrix for T times the coordinate matrix for x is the coordinate matrix for T(x).

2015/7/2 Elementary Linear Algebra 44 Example Let T : P 1  P 2 be the transformations defined by T (p(x)) = xp(x). Find the matrix for T with respect to the standard bases, B = {u 1, u 2 } and B = {v 1, v 2, v 3 }, where u 1 = 1, u 2 = x ; v 1 = 1, v 2 = x, v 3 = x 2 Solution:  T(u 1 ) = T(1) = (x)(1) = x and T(u 2 ) = T(x) = (x)(x) = x 2  [T (u 1 )] B ’ = [0 1 0] T [T (u 2 )] B ’ = [0 0 1] T  Thus, the matrix for T w.r.t. B and B ’ is

2015/7/2 Elementary Linear Algebra 45 Example Let T : R 2  R 3 be the linear transformation defined by Find the matrix for the transformation T with respect to the bases B = {u 1,u 2 } for R 2 and B = {v 1,v 2,v 3 } for R 3, where Solution:

2015/7/2 Elementary Linear Algebra 46 Example

2015/7/2 Elementary Linear Algebra 47 Theorems Theorem  If T : R n  R m is a linear transformation and if B and B are the standard bases for R n and R m, respectively, then [T] B,B = [T] Theorem  If T 1 : U  V and T 2 : V  W are linear transformations, and if B, B  and B are bases for U, V and W, respectively, then [T 2  T 1 ] B,B ’ = [T 2 ] B ’,B ’’ [T 1 ] B ’’,B

Theorem If T : V  V is a linear operator and if B is a basis for V then the following are equivalent  T is one to one  [T] B is invertible Moreover, when these equivalent conditions hold [T -1 ] B = [T] B /7/2 Elementary Linear Algebra 48

2015/7/2 Elementary Linear Algebra 49 Indirect Computation of a Linear Transformation An indirect procedure to compute a linear transformation: 1)Compute the coordinate matrix [x] B 2)Multiply [x] B on the left by [T] B,B to produce [T (x)] B 3)Reconstruct T (x) from its coordinate matrix [T (x)] B x T (x) [x]B[x]B [T (x)] B Direction computation Multiply by [T] B,B (1) (2) (3)

2015/7/2 Elementary Linear Algebra 50 Example Let T : P 2  P 2 be linear operator defined by T(p(x)) = p(3x – 5), that is, T (c o + c 1 x + c 2 x 2 ) = c o + c 1 (3x – 5) + c 2 (3x – 5) 2  Find [T] B with respect to the basis B = {1, x, x 2 }  Use the indirect procedure to compute T (1 + 2x + 3x 2 )  Check the result by computing T (1 + 2x + 3x 2 ) Solution:  Form the formula for T, T(1) = 1, T(x) = 3x – 5, T(x 2 ) = (3x – 5) 2 = 9x 2 – 30x + 25  Thus,

2015/7/2 Elementary Linear Algebra 51 Example  The coordinate matrix relative to B for vector p = 1 + 2x + 3x 2 is [p] B = [1 2 3] T.  Thus, [T (1 + 2x + 3x 2 )] B = [T (p)] B = [T] B [p] B =  T (1 + 2x + 3x 2 ) = 66 – 84x + 27x 2  By direction computation: T (1 + 2x + 3x 2 ) = 1 + 2(3x – 5) + 3(3x – 5) 2 = 1 + 6x – x 2 – 90x + 75 = 66 – 84x + 27x 2 x T (x) [x]B[x]B [T (x)] B Direction computation Multiply by [T] B,B (1) (2) (3)

2015/7/2 Elementary Linear Algebra 52 Chapter Content General Linear Transformations Kernel and Range Inverse Linear Transformations Matrices of General Linear Transformations Similarity Isomorphism

2015/7/2 Elementary Linear Algebra 53 Similarity The matrix of a linear operator T : V  V depends on the basis selected for V that makes the matrix for T as simple as possible – a diagonal or triangular matrix.

2015/7/2 Elementary Linear Algebra 54 Simple Matrices for Linear Operators Consider the linear operator T : R 2  R 2 defined by and the standard basis B = {e 1, e 2 } for R 2. The matrix for T with respect to this basis is the standard matrix for T; that is, [T] B = [T] = [T(e 1 ) | T(e 2 )]. Since T (e 1 ) = [1 -2] T, T (e 2 ) = [1 4] T, we have However, if u 1 = [1 1] T, u 2 = [1 2] T, then the matrix for T with respect to the basis B = {u 1, u 2 } is the diagonal matrix

2015/7/2 Elementary Linear Algebra 55 Theorem If B and B are bases for a finite-dimensional vector space V, and if I : V  V is the identity operator, then [I] B,B is the transition matrix from B to B. Remark V I vv Basis = B V [I] B,B is the transition matrix from B to B.

2015/7/2 Elementary Linear Algebra 56 Theorem Theorem  Let T : V  V be a linear operator on a finite-dimensional vector space V, and let B and B be bases for V. Then [T] B = P -1 [T] B P where P is the transition matrix from B to B. Remark: V IIT Basis = B VVV v v T(v)T(v)T(v)T(v) [T] B = [I] B,B [T] B [I] B,B = P -1 [T] B P

2015/7/2 Elementary Linear Algebra 57 Example Let T : R 2  R 2 be defined by Find the matrix T with respect to the standard basis B = {e 1, e 2 } for R 2, then use Theorem to find the matrix of T with respect to the basis B = {u 1, u 2 }, where u 1 = [1 1] T and u 2 = [1 2] T. Solution:

2015/7/2 Elementary Linear Algebra 58 Definitions Definition  If A and B are square matrices, we say that B is similar to A if there is an invertible matrix P such that B = P -1 AP Definition  A property of square matrices is said to be a similarity invariant or invariant under similarity if that property is shared by any two similar matrices.

2015/7/2 Elementary Linear Algebra 59 Similarity Invariants PropertyDescription DeterminantA and P -1 AP have the same determinant. InvertibilityA is invertible if and only if P -1 AP is invertible. RankA and P -1 AP have the same rank. NullityA and P -1 AP have the same nullity. TraceA and P -1 AP have the same trace. Characteristic polynomialA and P -1 AP have the same characteristic polynomial. EigenvaluesA and P -1 AP have the same eigenvalues Eigenspace dimension If is an eigenvalue of A and P -1 AP then the eigenspace of A corresponding to and the eigenspace of P -1 AP corresponding to have the same dimension.

2015/7/2 Elementary Linear Algebra 60 Determinant of A Linear Operator Two matrices representing the same linear operator T : V  V with respect to different bases are similar. For any two bases B and B we must have det([T] B ) = det([T] B ) Thus we define the determinant of the linear operator T to be det(T) = det([T] B ) where B is any basis for V. Example  Let T : R 2  R 2 be defined by

2015/7/2 Elementary Linear Algebra 61 Eigenvalues of a Linear Operator A scalar is called an eigenvalue of a linear operator T : V  V if there is a nonzero vector x in V such that Tx = x. The vector x is called an eigenvector of T corresponding to. Equivalently, the eigenvectors of T corresponding to are the nonzero vectors in the kernel of I – T. This kernel is called the eigenspace of T corresponding to.

Eigenvalues of a Linear Operator If V is a finite-dimensional vector space, and B is any basis for V, then  The eigenvalues of T are the same as the eigenvalues of [T] B.  A vector x is an eigenvector of T corresponding to [T] B if and only if its coordinate matrix [x] B is an eigenvector of [T] B corresponding to. 2015/7/2 Elementary Linear Algebra 62

2015/7/2 Elementary Linear Algebra 63 Example Find the eigenvalues and bases for the eigenvalues of the linear operator T : P 2  P 2 defined by T (a + bx + cx 2 ) = -2c + (a + 2b + c)x + (a + 3c)x 2 Solution:  The eigenvalues of T are  = 1 and  = 2  The eigenvectors of [T] B are:  = 2:  = 1:

2015/7/2 Elementary Linear Algebra 64 Example Let T : R 3  R 3 be the linear operator given by Find a basis for R 3 relative to which the matrix for T is diagonal. Solution:  det(    

2015/7/2 Elementary Linear Algebra 65 Onto Transformations Let V and W be real vector spaces. We say that the linear transformation T : V  W is onto if the range of T is W. An onto transformation is also said to be surjective or to be a surjection. For a surjective mapping, the range and the codomain coincide. If a transformation T : V  W is both one-to-one (also called injective or an injection) and onto, then it is a one-to-one mapping to its range W and so has an inverse T -1 : W  V. A transformation that is one-to-one and onto is also said to be bijective or to be a bijection between V and W.

Theorem Bijective Linear Transformation  Let V and W be finite-dimensional vector spaces. If dim(V)  dim(W), then there can be no bijective linear transformation from V to W. 2015/7/2 Elementary Linear Algebra 66

2015/7/2 Elementary Linear Algebra 67 Chapter Content General Linear Transformations Kernel and Range Inverse Linear Transformations Matrices of General Linear Transformations Similarity Isomorphism

2015/7/2 Elementary Linear Algebra 68 Isomorphisms Definition  An isomorphism between V and W is a bijective linear transformation from V to W.

Isomorphisms Theorem (Isomorphism Theorem)  Let V be a finite-dimensional real vector space. If dim(V) = n, then there is an isomorphism from V to R n. Example  The vector space P 3 is isomorphic to R 4, because the transformation T(a + bx + cx 2 + dx 3 ) = (a,b,c,d) is one-to-one, onto, and linear. 2015/7/2 Elementary Linear Algebra 69

2015/7/2 Elementary Linear Algebra 70 Isomorphisms between Vector Spaces Theorem (Isomorphism of Finite-Dimensional Vector Spaces)  Let V and W be finite-dimensional vector spaces. If dim(V) = dim(W), then V and W are isomorphic.

Example An Isomorphism between P 3 and M 22  Because dim(P 3 ) = 4 and dim(M 22 ) = 4, these spaces are isomorphic.  We can find an isomorphism T : P 3  M 22 :  This is one-to-one and onto linear transformation, so it is an isomorphism between P 3 and M /7/2 Elementary Linear Algebra 71