CPSC 491

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 …

3. Matrix-Vector Multiplication • Linear (the 1st 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

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

5. Linear Mapping • is a linear mapping, which satisfies • Distributive law • Associative law (for scalar) • Conversely, every linear map from Rn to Rmcan be expressed as a multiplication by anmxnmatrix

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

7. Mat-mat multiplication • Matrix-matrix multiplication • is defined as • We can calculate Bcolumnwisely • Each column of B is a linear combination of the columns aj with the coefficients ckj

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.

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

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

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 Rm • Any vector in Rm 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-1A = I • I is the mxm identity matrix • The inverse of a nonsingular matrix is unique. • A-1b is the unique solution of Ax = b. • A-1b is the vector of coefficients of the expansion of b in the basis of the columns of A.

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

13. Inner product • Inner product • Euclidean length • Angle

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

15. Orthonormal • Definition • Theorem • Corollary

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

17. Components of a vector

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

19. An example • 2D rotation matrix

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