First lecture on Linear Algebra

1. Math 15: Linear Algebra MATRIX and OPERATIONS ON MATRIX MATRIX - any rectangular array of m x n real numbers arranged in m horizontal rows and n vertical columns.

2. Math 15: Linear Algebra MATRIX and OPERATIONS ON MATRIX a ij – element found in the i th row and j th column size of the matrix – m x n

3. Math 15: Linear Algebra Types of Matrix: 1. SQUARE MATRIX - an n x n matrix, same number of rows and columns 2. DIAGONAL MATRIX - a square matrix in which a ij = 0 when i is not equal to j.

4. Math 15: Linear Algebra Illustrations:

5. Math 15: Linear Algebra Types of Matrix: 3. SCALAR MATRIX - diagonal matrix in which all terms in the main diagonal are equal.

6. Math 15: Linear Algebra Types of Matrix: 4. IDENTITY MATRIX - diagonal matrix in which all terms in the main diagonal are equal to 1.

8. Math 15: Linear Algebra Operations on Matrices: 2. Scalar Product: B = rA, where b ij = r x a ij and r is a scalar number.

9. Math 15: Linear Algebra Illustrations: Evaluate: 1. A + B 3. 2A+3B 2. A – B 4. 4A - B

10. Math 15: Linear Algebra Operations on Matrices: 3. Dot Product: C = A B, where c ij = a ij x b ij, provided A and B are of the same size. Illustration: Evaluate A B from the previous example.

11. Math 15: Linear Algebra Operations on Matrices: 4. MATRIX MULTIPLICATION C = AB, where C ij = a i1 b 1j + a i2 b 2j +… a ip b 1p size of A  m x p size of B  p x n size of C  m x n

12. Math 15: Linear Algebra Illustrations: Evaluate both AB and BA. What did you observe? Note: AB is not necessarily equal to BA.

13. Math 15: Linear Algebra Operations on Matrices: 5. Matrix Transpose, A T (or A’) If A is an m x n matrix, then n x m matrix A T = [a ij T ], where a ij T = a ji . Illustration: Evaluate A T and B T from the previous example.

14. Math 15: Linear Algebra Definition: A matrix A is symmetric if A T = A. Illustration: Verify if the following matrix is symmetric.

16. Math 15: Linear Algebra Properties of Matrix Operations: 2. Scalar Multiplication a. r(sA) = (rs)A b. (r+s)A = rA +sA c. r(A+ B) = rA + rB d. A(rB) = r(AB)

17. Math 15: Linear Algebra Properties of Matrix Operations: 3. Matrix Multiplication a. A(BC) = (AB)C b. A(B+C) = AB +AC c. (A+ B)C = AC + BC

18. Math 15: Linear Algebra Properties of Matrix Operations: 4. Matrix Transpose a. (A T ) T = A b. (A+B) T = A T +B T c. (A B) T = B T A T d. (rA) T = rA T

19. Math 15: Linear Algebra End