1 MAC 2103 Module 6 Euclidean Vector Spaces I

2 Rev.F09 Learning Objectives Upon completing this module, you should be able to: 1. Use vector notation in ℜ n. 2. Find the inner product of two vectors in ℜ n. 3. Find the norm of a vector and the distance between two vectors in ℜ n. 4. Express a linear system in ℜ n in dot product form. 5. Find the standard matrix of a linear transformation from ℜ n to ℜ m. 6. Use linear transformations such as reflections, projections, and rotations. 7. Use the composition of two or more linear transformations. Click link to download other modules.

3 Rev.09 Euclidean Vector Spaces I Click link to download other modules. Euclidean n-Space, ℜ n Linear Transformations from ℜ n to ℜ m There are two major topics in this module:

4 Rev.F09 Some Important Properties of Vector Operations in ℜ n Click link to download other modules. If u, v, and w are vectors in ℜ n and k and s are scalars, then the following hold: (See Theorem 4.1.1) a) u + v = v + u b) u + ( v + w ) = (u + v) + w c) u + 0 = 0 + u = ud) u + (-u) = 0 e) k(su) =(ks)uf) k(u + v) = ku + kv g) (k + s)u = ku + su h) 1u = u

5 Rev.F09 Basic Vector Operations in ℜ n Click link to download other modules. Two vectors u = (u 1, u 2, …, u n ) and v = (v 1, v 2, …, v n ) are equal if and only if u 1 = v 1, u 2 = v 2, …, u n = v n. Thus, u + v = (u 1 + v 1, u 2 + v 2,…, u n + v n ) u - v = (u 1 - v 1, u 2 - v 2,…, u n - v n ) and 5v - 2u = (5u 1 - 2v 1, 5u 2 - 2v 2,…, 5u n - 2v n )

6 Rev.F09 How to Find the Inner Product of Two Vectors in ℜ n ? Click link to download other modules. The inner product of two vectors u = (u 1,u 2,…,u n ) and v = (v 1,v 2,…,v n ), u · v, in ℜ n is also known as the Euclidean inner product or dot product. The inner product, u · v, can be computed as follows: Example: Find the Euclidean inner product of u and v in ℜ 4, if u = (2, -3, 6, 1) and v = (1, 9, -2, 4). Solution:

7 Rev.F09 How to Find the Norm of a Vector in ℜ n ? Click link to download other modules. As we have learned in a previous module, the norm of a vector in ℜ 2 and ℜ 3 can be obtained by taking the square root of the sum of square of the components as follows:

8 Rev.F09 How to Find the Norm of a Vector in ℜ n ? (Cont.) Click link to download other modules. Similarly, the Euclidean norm of u = (u 1,u 2,…,u n ), ||u||, in ℜ n can be computed as follows: Example: Find the Euclidean norm of u = (2, -3, 6, 1) in ℜ 4. Solution:

9 Rev.F09 How to Find the Distance Between Two Vectors in ℜ n ? Click link to download other modules. The distance between u = (u 1,u 2,…,u n ) and v = (v 1,v 2,…,v n ) in ℜ n, d(u,v), is also known as the Euclidean distance. The Euclidean distance, d(u,v), can be computed as follows: Example: Suppose u = (2, -3, 6, 1) and v = (1, 9, -2, 4). Find the Euclidean distance between u and v in ℜ 4, Solution:

10 Rev.F09 How to Express a Linear System in ℜ n in Dot Product Form? Click link to download other modules. Example: Express the following linear system in dot product form. Solution:

11 Rev.F09 How to Express a Linear Transformation from ℜ 3 to ℜ 4 in Matrix Form? Click link to download other modules. The linear transformation T: ℜ 3 → ℜ 4 defined by the equations can be expressed in matrix form as follows:

12 Rev.F09 What is the Standard Matrix for a Linear Transformation? Click link to download other modules. Based on our example in previous slide, the standard matrix can be found from the linear transformation T: ℜ 3 → ℜ 4 expressed in matrix form. The standard matrix for T is:

13 Rev.F09 Example and Notations Click link to download other modules. Example: Find the standard matrix for the linear transformation T defined by the formula as follows: Solution: In this case, the linear operator T assigns a unique point (w 1, w 2 ) in ℜ 2 to each point (x 1, x 2 ) in ℜ 2 according to the rule or as a linear system, it is as follows:Note: A linear transformation T: ℜ n → ℜ m is also known as a linear operator.

14 Rev.F09 Example and Notations (Cont.) Click link to download other modules. A linear system can be expressed in matrix form. In this case, the standard matrix for T is In general, the linear transformation is represented by T: ℜ n → ℜ m or T A : ℜ n → ℜ m ; the matrix A = [a ij ] is called the standard matrix for the linear transformation, and T is called multiplication by A.

15 Rev.F09 Zero Transformation and Identity Operator Click link to download other modules. If 0 is the m x n zero matrix, then for every vector x in ℜ n, we will have the zero transformation from ℜ n to ℜ m, T 0 : ℜ n → ℜ m, where T 0 is called multiplication by 0. If I is the n x n identity matrix, then for every vector x in ℜ n, we will have an identity operator on ℜ n, T I : ℜ n → ℜ n, where T I is called multiplication by I. Next, we will look at some important operators on ℜ 2 and ℜ 3, namely the linear operators that produce reflections, projections, and rotations.

16 Rev.F09 Linear Operators for Reflection Click link to download other modules. If the linear operator T: ℜ 2 → ℜ 2 maps each vector into its symmetric image about the y-axis, we can construct a reflection operator or linear transformation as follows: y (-x,y) (x,y) w u x

17 Rev.F09 Linear Operators for Reflection (Cont.) Click link to download other modules. If the linear operator T: ℜ 2 → ℜ 2 maps each vector into its symmetric image about the x-axis, we can construct a reflection operator or linear transformation as follows: y (x,y) u x w (x,-y)

18 Rev.F09 Linear Operators for Reflection (Cont.) Click link to download other modules. If the linear operator T: ℜ 2 → ℜ 2 maps each vector into its symmetric image about the line y = x, we can construct a reflection operator or linear transformation as follows: y (y,x) y = x w u (x,y) x

19 Rev.F09 Linear Operators for Reflection (Cont.) Click link to download other modules. If the linear operator T: ℜ 3 → ℜ 3 maps each vector into its symmetric image about the xy-plane, we can construct a reflection operator or linear transformation as follows: z u (x,y,z) y w (x,y,-z) x

20 Rev.F09 Orthogonal Projection Operator Click link to download other modules. If the linear operator T: ℜ 3 → ℜ 3 maps each vector into its orthogonal projection on the xy-plane, we can construct a projection operator or linear transformation as follows: z u (x,y,z) y w (x,y,0) x

21 Rev.F09 Orthogonal Projection Operator (Cont.) Click link to download other modules. Example: Use matrix multiplication to find the orthogonal projection of (-9,4,3) on the xy-plane. From previous slide, the standard matrix for the linear operator T mapping each vector into its orthogonal projection on the xy-plane in ℜ 3 is obtained: So the orthogonal projection, w, of (-9,4,3) on the xy-plane is: Thus, T(-9,4,3) = (-9,4,0).

22 Rev.F09 Linear Operators for Rotation Click link to download other modules. If the linear operator T: ℜ 2 → ℜ 2 rotates each vector counterclockwise in ℜ 2 through a fixed angle θ in ℜ 2, we can construct a rotation operator or linear transformation as follows: y (w 1,w 2 ) w θ u (x,y) ɸ x Hint: Let r = ||u||=||w||, then use x = r cos( ɸ ), y = r sin( ɸ ), w 1 =r cos(θ+ ɸ ), w 2 = r sin(θ+ ɸ ), and trigonometry identities.

23 Rev.F09 Linear Operators for Rotation (Cont.) Click link to download other modules. y (w 1,w 2 ) w θ u (x,y) ɸ x

24 Rev.F09 Linear Operators for Rotation (Cont.) Click link to download other modules. Example: Use matrix multiplication to find the image of the vector (3,-4) when it is rotated through an angle, θ, of 30°. Since the standard matrix for the linear operator T rotating each vector through an angle of θ (counterclockwise) in ℜ 2 has been obtained:

25 Rev.F09 Linear Operators for Rotation (Cont.) Click link to download other modules. It follows that the image, w, of (3,-4) when it is rotated through an angle of 30° (counterclockwise) in ℜ 2 can be found as: Thus,

26 Rev.F09 Composition of Linear Transformations Click link to download other modules. If T A : ℜ n → ℜ k and T B : ℜ k → ℜ m are linear transformations, then the application of T A followed by T B produces a transformation from ℜ n to ℜ m ; this transformation is called the composition of T B with T A and is denoted by T B ○ T A. The composition T B ○ T A is linear because Thus, T B ○ T A is multiplication by BA and can be expressed as T B ○ T A = T BA. Alternatively, we have [T B ○ T A ] = [ T B ][ T A ].

27 Rev.F09 Composition of Linear Transformations (Cont.) Click link to download other modules. Example: Find the standard matrix for the stated composition of linear operators on ℜ 2, if a rotation of π/2 is followed by a reflection about the line y = x. We know the standard matrix for the linear operator T A rotating each vector through an angle of θ = π/2 (counterclockwise) in ℜ 2 is as follows:

28 Rev.F09 Composition of Linear Transformations (Cont.) Click link to download other modules. We also know the standard matrix for the linear operator, T B, reflecting each vector about the line y = x in ℜ 2 is as follows: The composition we want is the linear operator T: T = T B ○ T A (rotation followed by reflection). Therefore, the standard matrix for T is [T] = [T B ○ T A ] = [ T B ][ T A ]. Note: This is the symmetric image about the x-axis matrix. See slide 17.

29 Rev.F09 What have we learned? We have learned to: 1. Use vector notation in ℜ n. 2. Find the inner product of two vectors in ℜ n. 3. Find the norm of a vector and the distance between two vectors in ℜ n. 4. Express a linear system in ℜ n in dot product form. 5. Find the standard matrix of a linear transformation from ℜ n to ℜ m. 6. Use linear transformations such as reflections, projections, and rotations. 7. Use the composition of two or more linear transformations. Click link to download other modules.

30 Rev.F09 Credit Some of these slides have been adapted/modified in part/whole from the following textbook: Anton, Howard: Elementary Linear Algebra with Applications, 9th Edition Click link to download other modules.