1. Chapter 4 – Linear Spaces
Outline Introduction to Linear Spaces Linear Transformations and Isomorphisms The Matrix of a Linear Transformation

2. 4.1 Introduction to Linear Spaces
(Definition 4.1.1) A linear space V is a set endowed with a rule for addition (if f and g are in V, then so is f+g) and a rule for scalar multiplication (if f is in V and k in R, then kf is in V) such that these operations satisfy the following 8 rules (for all f, g, h in V and all c,k in R)
(f+g)+h=f+(g+h)
f+g=g+f
There is a neutral element n in V such that f+n=f, for all f in V. This n is unique and denoted by 0.
For each f in V there is a g in V such that f+g=0. This g is unique and denoted by (-f)
k(f+g)=kf+kg
(c+k)f=cf+kf
c(kf)=(ck)f
1f=f

3. Example 1
(Example 1) Consider the differential equation (DE) f’’(x)+f(x)=0, or f’’(x)=-f(x). Find the solutions of the DE.

4. Example 2
(Example 2) In Rn, the prototype linear space, the neutral element is the zero vector,

5. Example 3
(Example 3) Let F(R, R) be the set of all functions from R to R (see Example 1), with the operations (f+g)(x)=f(x)+g(x) and (kf)(x)=kf(x). Then F(R, R) is a linear space. The neutral element is the zero function, f(x)=0 for all x.

6. Example 4
(Example 4) If addition and scalar multiplication are given as in Definition 1.3.5, then Rn×m, the set of all n×m matrices, is a linear space. The neutral element is the zero matrix, whose entries are all zero.

7. Example 5
(Example 5) The set of all infinite sequences of real numbers is a linear space, where addition and scalar multiplication are defined term by term: (x0, x1, x2,…)+(y0, y1, y2,…)=(x0+y0, x1+y1, x2+y2,…) k(x0, x1, x2,…)=(kx0, kx1, kx2,…). The neutral element is the sequence (0,0,0,…).

8. Example 6
(Example 6) The linear equations in three unknowns, ax+by+cz=d where a, b, c, and d are constants, form a linear space. The operations (addition and scalar multiplication) are familiar from the process of Gaussian elimination discussed in Chapter 1. The neutral element is the equation 0=0 (with a=b=c=d=0).

9. Example 7
(Example 7) Consider the plane with a point designated as the origin, O, but without a coordinate system (the coordinate-free plane). A geometric vector in this plane is an arrow (a directed line segment) with its tail at the origin. The sum of two vectors and is defined by means of a parallelogram, but is k times as long as . If k is negative, then points in the opposite direction, and it is |k| times as long as The geometric vectors in the plane with these operations forms a linear space. The neutral element is the zero vector , with tail and head at the origin.

10. Example 8
(Example 8) Let C be the set of the complex numbers. We trust that you have at least a fleeting acquaintance with complex numbers. Without attempting a definition, we recall that a complex number can be expressed as z=a+bi, where a and b are real numbers. Addition of complex numbers is defined in a natural way, by the rule (a+ib)+(c+id)=(a+c)+i(b+d). If k is a real scalar, we define k(a+ib)=ka+i(kb). There is also a (less natural) rule for the multiplication of complex numbers, but we are not concerned with this operations just given form a linear space; the neutral element is the complex number 0=0+0i.

11. Example 9
Let Show that is a linear combination of A and I2.

12. (Definition 4.1.2) Subspaces
(Definition 4.1.2) A subset W of a linear space V is called a subspace of V if
W contains the neutral element 0 of V.
W is closed under addition (if f and g are in W, then so is f+g).
W is closed under scalar multiplication (if f is in W and k is a scalar, then kf is in W).
W is closed under linear combinations.

13. Example 10
Show that the polynomials of degree≦2, of the form f(x)=a+bx+cx2, are a subspace W of the space F(R, R) of all functions from R to R.

14. Example 11
(Example 11) Show that the differentiable functions from a subspace W of F(R, R).

15. (Example 12) Here are more subspace of F(R, R):
C∞, the smooth functions, that is, functions we can differentiate as many times as we want. This subspace contains all polynomials, exponential functions, sin(x), and cos(x), for example.
P, the set of all polynomials.
Pn, the set of all polynomials of degree≦n.

16. Example 13
Show that the matrices B that commute with form a subspace of R2×2.

17. Example 14
(Example 14) Consider the set W of all noninvertible 2×2 matrices. Is W a subspace of R2×2?

18. (Definition 4.1.3) Span, Linear Independent, Basis, Coordinates
(Definition 4.1.3) Consider the elements f1, f2,…,fn of a linear space V.
We say that f1, f2,…,fn span V if every f in V can be expressed as a linear combination of f1, f2,…,fn.
We say f1, f2,…,fn are (linear) independent if the equation c1f1+c2f2+…+cnfn=0 has only the trivial solution c1=c2=…=cn=0.
We say that elements f1, f2,…,fn are a basis of V if they span V and are independent. This means that every f in V can be written uniquely as a linear combination f= c1f1+c2f2+…+cnfn The coefficients c1, c2,…,cn are called the coordinates of f with respect to the basis B=(f1, f2,…,fn ). The vector in Rn is called the B-coordinate vector of f, denoted by [f]B.
The transformation is called the B-coordinate transformation.

19. (Fact 4.1.4) Linearity of Coordinates
(Fact 4.1.4) If B is a basis of a linear space V, then
[f+g]B=[f]B+[g]B, for all elements f and g of V, and
[kf]B=k[f]B, for all f in V and for all scalars k.

20. (Fact 4.1.5) Dimension
(Fact 4.1.5) If a linear space V has a basis with n elements, then all other bases of V consist of n elements as well. We say that n is the dimension of V: dim(V)=n.

21. Example 15
(Example 15) Find a basis of R2×2, the space of all 2×2 matrices, and thus determine the dimension of R2×2.

22. Example 16
(Example 16) Find a basis of P2, the space of all polynomials of degree≦2, and thus determine the dimension of P2.

23. Summary 4.1.6 Finding a Basis of a Linear Space V
Write down a typical element of V in terms of some arbitrary constants.
Using the arbitrary constants as coefficients, express your typical element as a linear combination.
Verify that the elements of V in this linear combination are linearly independent; then, they form a basis of V.

24. Example 17
(Example 17) Find a basis of the space V of all matrices B that commute with (See Example 13.)

25. (Fact 4.1.7) Linear Differential Equations
(Fact 4.1.7) The solutions of the DE f”(x)+af’(x)+bf(x)=0 (where a and b are constants) form a two-dimensional subspace of the space C∞ of smooth functions. More generally, the solutions of the DE f(n)(x)+an-1f(n-1)(x)+…+a1f’(x)+a0f(x)= (where the ai are constants) form an n-dimensional subspace of C∞. A DE of this form is called an nth-order linear differential equation.

26. Example 18
(Example 18) Find all solutions of the DE f”(x)+f’(x)-6f(x)=0. (Hint: Find all exponential functions f(x)=ekx that solve the DE.)

27. Example 19
Let f1,…,fn be polynomials. Explain why these polynomials will not span the space P of all polynomials.

28. (Definition 4.1.8) Finite dimension linear spaces
(Definition 4.1.8) A linear space V is called finite-dimensional if it has a (finite) basis f1, f2,…,fn, so that we can define its dimension dim(V)=n. (See Definition 4.1.5) Otherwise the space is called infinite-dimensional.

29. 4.2 Linear Transformations and Isomorphisms
(Definition 4.2.1) Linear transformations, image, kernel, rank, nullity. Consider two linear spaces V and W. A function T from V to W is called a linear transformation if T(f+g)=T(f)+T(g) and T(kf)=kT(f) for all elements f and g of V and for all scalars k. for a linear transformation T from V to W, we let im(T)={T(f):f in V} and ker(T)={f in V: T(f)=0} Note that im(T) is a subspace of codomain W and that ker(T) is a subspace of domain V. If the image of T is finite-dimensional, then dim(im T) is called the rank of T, and if the kernel of T is finite-dimensional, then dim(ker T) is the nullity of T. If V is finite-dimensional, then the rank-nullity theorem holds: dim(V)=rank(T)+nullity(T)=dim(im T)+dim(ker T)

30. Example 1
(Example 1) Consider the transformation D(f)=f’ from C∞ to C ∞. It follows from the rules of calculus that D is a linear transformation: D(f+g)=(f+g)’=f’+g’ equals D(f)+D(g)=f’+g’ and D(kf)=(kf)’=kf’ equals kD(f)=kf’. Here f and g are smooth functions, and k is a constant. What is the kernel of D? What about the image of D?

31. Example 2
(Example 2) Let C[0,1] be the linear space of all continuous functions from the closed interval [0,1] to R. We define the transformation Is the transformation I linear? What is the image of I? What is the kernel of I?

32. Example 3
(Example 3) Let V be the space of all infinite sequences of real numbers. Consider the transformation T(x0, x1, x2,…)=(x1, x2, x3,…) from V to V. (We drop the first term, x0, of the sequence.)
Show that T is a linear transformation.
Find the kernel of T.
Is the sequence (1, 2, 3,…) in the image of T.
Find the image of T.

33. Example 4
(Example 4) Consider the transformation Note that T is the coordinate transformation with respect to the standard basis

34. (Definition 4.2.2) Isomorphisms and Isomorphic Space
An invertible linear transformation is called an isomorphism. We say that the linear spaces V is isomorphic to the linear space W if there is an isomorphism from V to W.

35. (Fact 4.2.3) Coordinate Transformations are Isomorphisms
(Fact 4.2.3) If B=(f1, f2,…,fn) is a basis of a linear space V, then the coordinate transformation T(f)=[f]B from V to Rn is an isomorphism. Thus V is isomorphic to Rn; the linear spaces V and Rn have the same structure.

36. Example 5
(Example 5) Show that the transformation T(A)=S-1AS from R2×2 to R2×2 is an isomorphism, where

37. (Fact 4.2.4) Properties of Isomorphisms
(a) A linear transformation T from V to W is an isomorphism if (and only if) ker(T)={0} and im(T)=W. In parts (b) and (c), the linear space V and W are assumed to be finite dimensional.
(b) If V is isomorphic to W, then dim(V)=dim(W).
(c) Suppose T is a linear transformation from V to W with ker(T)={0}. If dim(V)=dim(W), then T is an isomorphism.

38. Example 6
(Example 6) (a) Is the linear transformation from P3 to R3 an isomorphism? (b) Is the linear transformation from P2 to R3 an isomorphism?

39. 4.3 The Matrix of a Linear Transformation
(Example 1) Consider the linear transformation T(f)=f’+f” from P2 to P2, or, written more explicitly, T(f(x))=f’(x)+f”(x). Since P2 is isomorphic to R3, this is essentially a linear transformation from R3 to R3, given by a 3×3 matrix B.

40. (Fact 4.3.1) The B-matrix of a Linear Transformation
(Fact 4.3.1) Consider a linear transformation T from V to V, where V is an n-dimensional linear space. Let B be a basis of V. Then there exist one (and only one) n×n matrix B that transforms [f]B into [T(f)]B, called the B-matrix of T:

41. (Fact 4.3.2) The Columns of the B-matrix of a Linear Transformation
(Fact 4.3.2) Consider a linear transformation T from V to V, and let B be the matrix of T with respect to a basis B=(f1,…,fn) of V. Then The columns of B are the B-coordinate vectors of the transforms of the basis elements f1,…,fn of V.

42. Example 2
(Example 2) Use Fact to find the matrix B of the linear transformation T(f)=f’+f” from P2 to P2 with respect to the standard basis B=(1,x,x2); see Example 1.

43. Example 3
(Example 3) Let V be the span of cos(x) and sin(x) in C∞; note that V consists of all trigonomentric functions of the form f(x)=acos(x)+bsin(x). Consider the transformation T(f)=3f+2f’-f” from V to V. We are told that T is a linear transformation. a. Using Fact find the matrix B of T with respect tot the basis B=(cos(x), sin(x)). b. Is T an isomorphism?

44. Example 4
(Example 4) Consider the linear transformation a. Find the matrix B of T with respect to the standard basis B of R2×2. b. Find bases of the image and kernel of B. c. Find bases of the image and kernel of T, and thus determine rank and nullity of transformation T.

45. (Fact 4.3.3) Change of Basis Matrix
(Fact 4.3.3) Suppose that U and B are two bases of an n-dimensional linear space V. Then there exists one (and only one) n×m matrix S such that [f]U=S[f]B for all f in V. This invertible matrix S is called the change of basis matrix from B to U; it is often denoted by SB→U. If B=(f1,…,fn), then SB→U=[[f1]U … [fn]U], that is, the columns of SB→U are the U-coordinate vectors of the elements of basis B.

46. Example 5
(Example 5) Let V be the subspace of C∞ spanned by the functions ex and e-x, with the bases U=(ex, e-x) and B=(ex+e-x, ex-e-x). Find the change of basis matrix SB→U.

47. Example 6
(Example 6) Let be the standard basis of Rn, and let be an arbitrary basis of Rn. Find the change of basis matrix SB→U.

48. Example 7
(Example 7) The x1+x2+x3=0 defines a plane V in R3. In this plane, consider the two bases and a. Find the change of basis matrix S from B to U. b. Verify that the matrix S in part (a) satisfies the equation

49. (Fact 4.3.4) Change of Basis in a Subspace of Rn
(Fact 4.3.4) Consider a subspace V of Rn with two bases and Let S be the change of basis matrix from B to U. Then the following equation holds:

50. (Fact 4.3.5) Change of Basis for the Matrix of a Linear Transformation
(Fact 4.3.5) Let V be a linear space with two given bases U and B. Consider a linear transformation T from V to V, and let A and B be the U- and the B-matrix of T, respectively. Let S be the change of basis matrix from B to U. then A is similar to B, and AS=SB or A=SBS-1 or B=S-1AS.

51. Example 8
(Example 8) As in Example 5, let V be the linear space spanned by the functions ex and e-x, with the bases U=(ex, e-x) and B=(ex+e-x, ex-e-x). Consider the linear transformation D(f)=f’ from V to V. a. Find the U-matrix A of D. b. Use part (a), Fact 4.3.5, and Example 5 to find the B-matrix B of D. c. Use Fact to find the B-matrix B of D in terms of its columns.