1. 1 MAC 2103 Module 6 Euclidean Vector Spaces I

2. 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. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

3. 3 Rev.09 Euclidean Vector Spaces I http://faculty.valenciacc.edu/ashaw/ 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. 4 Rev.F09 Some Important Properties of Vector Operations in ℜ n http://faculty.valenciacc.edu/ashaw/ 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. 5 Rev.F09 Basic Vector Operations in ℜ n http://faculty.valenciacc.edu/ashaw/ 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. 6 Rev.F09 How to Find the Inner Product of Two Vectors in ℜ n ? http://faculty.valenciacc.edu/ashaw/ 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. 7 Rev.F09 How to Find the Norm of a Vector in ℜ n ? http://faculty.valenciacc.edu/ashaw/ 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. 8 Rev.F09 How to Find the Norm of a Vector in ℜ n ? (Cont.) http://faculty.valenciacc.edu/ashaw/ 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. 9 Rev.F09 How to Find the Distance Between Two Vectors in ℜ n ? http://faculty.valenciacc.edu/ashaw/ 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. 10 Rev.F09 How to Express a Linear System in ℜ n in Dot Product Form? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Example: Express the following linear system in dot product form. Solution:

11. 11 Rev.F09 How to Express a Linear Transformation from ℜ 3 to ℜ 4 in Matrix Form? http://faculty.valenciacc.edu/ashaw/ 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. 12 Rev.F09 What is the Standard Matrix for a Linear Transformation? http://faculty.valenciacc.edu/ashaw/ 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. 13 Rev.F09 Example and Notations http://faculty.valenciacc.edu/ashaw/ 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. 14 Rev.F09 Example and Notations (Cont.) http://faculty.valenciacc.edu/ashaw/ 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. 15 Rev.F09 Zero Transformation and Identity Operator http://faculty.valenciacc.edu/ashaw/ 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. 16 Rev.F09 Linear Operators for Reflection http://faculty.valenciacc.edu/ashaw/ 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. 17 Rev.F09 Linear Operators for Reflection (Cont.) http://faculty.valenciacc.edu/ashaw/ 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. 18 Rev.F09 Linear Operators for Reflection (Cont.) http://faculty.valenciacc.edu/ashaw/ 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. 19 Rev.F09 Linear Operators for Reflection (Cont.) http://faculty.valenciacc.edu/ashaw/ 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. 20 Rev.F09 Orthogonal Projection Operator http://faculty.valenciacc.edu/ashaw/ 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. 21 Rev.F09 Orthogonal Projection Operator (Cont.) http://faculty.valenciacc.edu/ashaw/ 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. 22 Rev.F09 Linear Operators for Rotation http://faculty.valenciacc.edu/ashaw/ 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. 23 Rev.F09 Linear Operators for Rotation (Cont.) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. y (w 1,w 2 ) w θ u (x,y) ɸ x

24. 24 Rev.F09 Linear Operators for Rotation (Cont.) http://faculty.valenciacc.edu/ashaw/ 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. 25 Rev.F09 Linear Operators for Rotation (Cont.) http://faculty.valenciacc.edu/ashaw/ 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. 26 Rev.F09 Composition of Linear Transformations http://faculty.valenciacc.edu/ashaw/ 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. 27 Rev.F09 Composition of Linear Transformations (Cont.) http://faculty.valenciacc.edu/ashaw/ 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. 28 Rev.F09 Composition of Linear Transformations (Cont.) http://faculty.valenciacc.edu/ashaw/ 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. 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. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

30. 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 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.