Lecture 9 Vector & Inner Product Space Last Time - Spanning Sets and Linear Independence (Cont.) - Basis and Dimension - Rank of a Matrix and Systems of Linear Equations - Midterm Exam Elementary Linear Algebra R. Larsen et al. (5 Edition) TKUEE翁慶昌-NTUEE SCC_11_2007

Lecture 9: Vector & Inner Product Spaces Today Applications of Vector Space Coordinates and Change of Basis Length and Dot Product in Rn Inner Product Spaces Reading Assignment: Secs 4.7,4.8, 5.1,5.2 Next Time Orthonormal Bases:Gram-Schmidt Process Mathematical Models and Least Square Analysis Applications Reading Assignment: Secs 5.3-5.5

What Have You Actually Learned about Linear Algebra So Far?

Today Applications of Vector Space Coordinates and Change of Basis Length and Dot Product in Rn Inner Product Spaces

Applications Null Space and Feasible Search Matrix Rank and Nullity.doc Industry Robot

4.7 Coordinates and Change of Basis Coordinate representation relative to a basis Let B = {v1, v2, …, vn} be an ordered basis for a vector space V and let x be a vector in V such that The scalars c1, c2, …, cn are called the coordinates of x relative to the basis B. The coordinate matrix (or coordinate vector) of x relative to B is the column matrix in Rn whose components are the coordinates of x.

Ex 1: (Coordinates and components in Rn) Find the coordinate matrix of x = (–2, 1, 3) in R3 relative to the standard basis S = {(1, 0, 0), ( 0, 1, 0), (0, 0, 1)} Sol:

Ex 3: (Finding a coordinate matrix relative to a nonstandard basis) Find the coordinate matrix of x=(1, 2, –1) in R3 relative to the (nonstandard) basis B ' = {u1, u2, u3}={(1, 0, 1), (0, – 1, 2), (2, 3, – 5)} Sol:

Change of basis problem: You were given the coordinates of a vector relative to one basis B and were asked to find the coordinates relative to another basis B'. Ex: (Change of basis) Consider two bases for a vector space V

Let

Transition matrix from B' to B: If [v]B is the coordinate matrix of v relative to B [v]B‘ is the coordinate matrix of v relative to B' where is called the transition matrix from B' to B

Thm 4.20: (The inverse of a transition matrix) If P is the transition matrix from a basis B' to a basis B in Rn, then (1) P is invertible (2) The transition matrix from B to B' is P–1 Notes:

Thm 4.21: (Transition matrix from B to B') Let B={v1, v2, … , vn} and B' ={u1, u2, … , un} be two bases for Rn. Then the transition matrix P–1 from B to B' can be found by using Gauss-Jordan elimination on the n×2n matrix as follows.

Ex 5: (Finding a transition matrix) B={(–3, 2), (4,–2)} and B' ={(–1, 2), (2,–2)} are two bases for R2 (a) Find the transition matrix from B' to B. (b) (c) Find the transition matrix from B to B' .

(the transition matrix from B' to B) Sol: (a) G.J.E. B B' I P-1 [I2 : P] = (the transition matrix from B' to B) (b)

(the transition matrix from B to B') (c) G.J.E. B' B I P-1 (the transition matrix from B to B') Check:

Rotation of the Coordinate Axes

Ex 6: (Coordinate representation in P3(x)) Find the coordinate matrix of p = 3x3-2x2+4 relative to the standard basis in P3(x), S = {1, 1+x, 1+ x2, 1+ x3}. Sol: p = 3(1) + 0(1+x) + (–2)(1+x2 ) + 3(1+x3 ) [p]s =

Ex: (Coordinate representation in M2x2) Find the coordinate matrix of x = relative to the standardbasis in M2x2. B = Sol:

Keywords in Section 4.7: coordinates of x relative to B：x相對於B的座標 coordinate matrix：座標矩陣 coordinate vector：座標向量 change of basis problem：基底變換問題 transition matrix from B' to B：從 B' 到 B的轉移矩陣

Today Applications of Vector Space Coordinates and Change of Basis Length and Dot Product in Rn Inner Product Spaces

5.1 Length and Dot Product in Rn The length of a vector in Rn is given by Notes: The length of a vector is also called its norm. Notes: Properties of length is called a unit vector.

Ex 1: (a) In R5, the length of is given by (b) In R3 the length of is given by (v is a unit vector)

A standard unit vector in Rn: Ex: the standard unit vector in R2: the standard unit vector in R3: Notes: (Two nonzero vectors are parallel) u and v have the same direction u and v have the opposite direction

Thm 5.1: (Length of a scalar multiple) Let v be a vector in Rn and c be a scalar. Then Pf:

Thm 5.2: (Unit vector in the direction of v) If v is a nonzero vector in Rn, then the vector has length 1 and has the same direction as v. This vector u is called the unit vector in the direction of v. Pf: v is nonzero (u has the same direction as v) (u has length 1 )

Notes: (1) The vector is called the unit vector in the direction of v. (2) The process of finding the unit vector in the direction of v is called normalizing the vector v.

Ex 2: (Finding a unit vector) Find the unit vector in the direction of , and verify that this vector has length 1. Sol: is a unit vector.

Distance between two vectors: The distance between two vectors u and v in Rn is Notes: (Properties of distance) (1) (2) if and only if (3)

Ex 3: (Finding the distance between two vectors) The distance between u=(0, 2, 2) and v=(2, 0, 1) is

Dot product in Rn: The dot product of and is the scalar quantity Ex 4: (Finding the dot product of two vectors) The dot product of u=(1, 2, 0, -3) and v=(3, -2, 4, 2) is

Thm 5.3: (Properties of the dot product) If u, v, and w are vectors in Rn and c is a scalar, then the following properties are true. (1) (2) (3) (4) (5) , and if and only if

Euclidean n-space: Rn was defined to be the set of all order n-tuples of real numbers. When Rn is combined with the standard operations of vector addition, scalar multiplication, vector length, and the dot product, the resulting vector space is called Euclidean n-space.

Ex 5: (Finding dot products) (b) (c) (d) (e) Sol:

Ex 6: (Using the properties of the dot product) Given Find Sol:

Thm 5.4: (The Cauchy - Schwarz inequality) If u and v are vectors in Rn, then ( denotes the absolute value of ) Ex 7: (An example of the Cauchy - Schwarz inequality) Verify the Cauchy - Schwarz inequality for u=(1, -1, 3) and v=(2, 0, -1) Sol:

The angle between two vectors in Rn: Same direction Opposite direction Note: The angle between the zero vector and another vector is not defined.

Ex 8: (Finding the angle between two vectors) Sol: u and v have opposite directions.

Orthogonal vectors: Two vectors u and v in Rn are orthogonal if Note: The vector 0 is said to be orthogonal to every vector.

Ex 10: (Finding orthogonal vectors) Determine all vectors in Rn that are orthogonal to u=(4, 2). Sol: Let

Thm 5.5: (The triangle inequality) If u and v are vectors in Rn, then Pf: Note: Equality occurs in the triangle inequality if and only if the vectors u and v have the same direction.

Thm 5.6: (The Pythagorean theorem) If u and v are vectors in Rn, then u and v are orthogonal if and only if

Dot product and matrix multiplication: (A vector in Rn is represented as an n×1 column matrix)

Keywords in Section 5.1: length: 長度 norm: 範數 unit vector: 單位向量 standard unit vector : 標準單位向量 normalizing: 單範化 distance: 距離 dot product: 點積 Euclidean n-space: 歐基里德n維空間 Cauchy-Schwarz inequality: 科西-舒瓦茲不等式 angle: 夾角 triangle inequality: 三角不等式 Pythagorean theorem: 畢氏定理

Today Applications of Vector Space Coordinates and Change of Basis Length and Dot Product in Rn Inner Product Spaces

5.2 Inner Product Spaces Inner product: Let u, v, and w be vectors in a vector space V, and let c be any scalar. An inner product on V is a function that associates a real number <u, v> with each pair of vectors u and v and satisfies the following axioms. (1) (2) (3) (4) and if and only if

Note: Note: A vector space V with an inner product is called an inner product space. Vector space: Inner product space:

Ex 1: (The Euclidean inner product for Rn) Show that the dot product in Rn satisfies the four axioms of an inner product. Sol: By Theorem 5.3, this dot product satisfies the required four axioms. Thus it is an inner product on Rn.

Ex 2: (A different inner product for Rn) Show that the function defines an inner product on R2, where and . Sol:

Note: (An inner product on Rn)

Ex 3: (A function that is not an inner product) Show that the following function is not an inner product on R3. Sol: Let Axiom 4 is not satisfied. Thus this function is not an inner product on R3.

Thm 5.7: (Properties of inner products) Let u, v, and w be vectors in an inner product space V, and let c be any real number. (1) (2) (3) Norm (length) of u: Note:

Distance between u and v: Angle between two nonzero vectors u and v: Orthogonal: u and v are orthogonal if .

Notes: (1) If , then v is called a unit vector. (2) (the unit vector in the direction of v) not a unit vector

Ex 6: (Finding inner product) is an inner product Sol:

Properties of norm: (1) (2) if and only if (3) Properties of distance: (1) (2) if and only if (3)

Thm 5.8： Let u and v be vectors in an inner product space V. (1) Cauchy-Schwarz inequality: (2) Triangle inequality: (3) Pythagorean theorem : u and v are orthogonal if and only if Theorem 5.4 Theorem 5.5 Theorem 5.6

Orthogonal projections in inner product spaces: Let u and v be two vectors in an inner product space V, such that . Then the orthogonal projection of u onto v is given by Note: If v is a init vector, then . The formula for the orthogonal projection of u onto v takes the following simpler form.

Ex 10: (Finding an orthogonal projection in R3) Use the Euclidean inner product in R3 to find the orthogonal projection of u=(6, 2, 4) onto v=(1, 2, 0). Sol:

Thm 5.9: (Orthogonal projection and distance) Let u and v be two vectors in an inner product space V, such that . Then

Keywords in Section 5.2: inner product: 內積 inner product space: 內積空間 norm: 範數 distance: 距離 angle: 夾角 orthogonal: 正交 unit vector: 單位向量 normalizing: 單範化 Cauchy – Schwarz inequality: 科西 - 舒瓦茲不等式 triangle inequality: 三角不等式 Pythagorean theorem: 畢氏定理 orthogonal projection: 正交投影