1. CPSC 491 Xin Liu Nov 17, 2010

2. Introduction Xin Liu PhD student of Dr. Rokne Contact liuxin@ucalgary.ca Slides downloadable at pages.cpsc.ucalgary.ca/~liuxin The way to math world Lecture attendance Hard to learn by yourselves Practices, practices, and practices … 2

3. Matrix-Vector Multiplication Linear (the 1 st degree) systems are the simplest, but most widely used systems in science and engineering A basic problem: solving the linear equation system Straight forward method Gaussian elimination Hard to do because large scale poor conditioned small disturbance in coefficients causes big difference in solutions A better method SVD – Singular Vale Decomposition Will be introduced gradually in a series of lectures 3

4. Definitions An n-vector is defined as Think about 3-vectors in Euclidean space An mxn matrix is defined as Multiplication 4

5. Linear Mapping is a linear mapping, which satisfies Distributive law Associative law (for scalar) Conversely, every linear map from R n to R m can be expressed as a multiplication by an mxn matrix 5

6. Mat-vect multiplication View the matrix-vector multiplication from another angle If we write A as a combination of column vectors Then the mat-vect multiplication can be written as That means: b is a linear combination of the columns of A 6

7. Mat-mat multiplication Matrix-matrix multiplication is defined as We can calculate B columnwisely Each column of B is a linear combination of the columns a j with the coefficients c kj 7

8. Range Definition: The range of a matrix A, is the set of vectors that can be expressed as Ax for some x. Theorem range (A) is the space spanned by the columns of A. The range of A is also called the column space of A. 8

9. Nullspace Definition: The nullspace (solution space) of A is the set of vectors x that satisfy Ax = 0. Each vector x in the nullspace gives the expansion coefficients of the zero vector as a linear combination of columns of A 9

10. Rank Column rank = dimension of space spanned by the matrix’s columns = # of linearly independent columns Row rank = dimension of space spanned by the matrix’s rows = # of linearly independent rows Row rank = Column rank = Matrix rank Full rank Theorem 10

11. Inverse A nonsingular or invertible matrix must be square and full rank. The m columns of a nonsingular mxm matrix A span (form a basis) for the whole space R m Any vector in R m can be expressed as a linear combination of the columns of A The inverse of A is a matrix A -1, such that AA -1 = A -1 A = I I is the mxm identity matrix The inverse of a nonsingular matrix is unique. A -1 b is the unique solution of Ax = b. A -1 b is the vector of coefficients of the expansion of b in the basis of the columns of A. 11

12. Transpose Definition The transpose A T of an mxn matrix A is nxm where the (i,j) entry of A T is the (j, i) entry of A. Example A is symmetric if A = A T. Multiplication 12

13. Inner product Euclidean length Angle 13

14. Orthogonal vectors Orthogonal (perpendicular) vectors Vectors x, y are orthogonal if x T y = 0. Orthogonal vector set Orthogonal two vector sets 14

15. Orthonormal Definition Theorem Corollary 15

16. Components of a vector Inner products can be used to decompose arbitrary vectors into orthogonal components (project onto orthonormal vectors). 16

17. Components of a vector 17

18. Orthogonal matrices Definition: According to the definition Or 18

19. An example 2D rotation matrix 19

20. Multiplication by an orthogonal matrix inner products is preserved angles between vectors are preserved lengths are preserved 20