Math15 Lecture1

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First lecture on Linear Algebra

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Math15 Lecture1

  1. 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. 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. 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. 4. Math 15: Linear Algebra Illustrations:
  5. 5. Math 15: Linear Algebra Types of Matrix: 3. SCALAR MATRIX - diagonal matrix in which all terms in the main diagonal are equal.
  6. 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.
  7. 7. Math 15: Linear Algebra Operations on Matrices: <ul><li>Matrix Addition: </li></ul><ul><li>C = A + B, where c ij = a ij + b ij, provided A and B are of the same size. </li></ul>Note: For matrix subtraction, C = A – B implies c ij = a ij - b ij .
  8. 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. 9. Math 15: Linear Algebra Illustrations: Evaluate: 1. A + B 3. 2A+3B 2. A – B 4. 4A - B
  10. 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. 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. 12. Math 15: Linear Algebra Illustrations: Evaluate both AB and BA. What did you observe? Note: AB is not necessarily equal to BA.
  13. 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. 14. Math 15: Linear Algebra Definition: A matrix A is symmetric if A T = A. Illustration: Verify if the following matrix is symmetric.
  15. 15. Math 15: Linear Algebra Properties of Matrix Operations: <ul><li>Addition </li></ul><ul><li>a. A + B = B + A </li></ul><ul><li>b. A + (B + C) = (A +B) + C </li></ul><ul><li>c. A+ 0 = A </li></ul><ul><li>d. A + (-A) = 0 </li></ul>
  16. 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. 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. 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. 19. Math 15: Linear Algebra End

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