Transposes n Permutations Multiplication by a permutation matrix P swaps the rows of a matrix when applying the method of elimination we use permutation matrices to move zeros out of piv ID: 198486
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� Transposes, n In this lecture we introduce vector spaces and their subspaces. Permutations Multiplication by a permutation matrix P swaps the rows of a matrix; when applying the method of elimination we use permutation matrices to move zeros out of pivot positions. 1 = P T , i.e. that P T P = I. Transposes When we 13 124 4 23 5 = . 331 41 A matrix A is symmetric if A T = A. Given any matrix R (not necessarily square) the product R T R is always symmetric, because � R R T = R T � R T T = R T R. (Note that � R T T = R.) Vector spaces We can add vectors and multiply b the origin to the point (a, b) which is a units to the right of the origin and b units above it, and we call R 2 the “x � y plane”. Another example of a space is R n , the set of (column) vectors MIT OpenCourseWare http://ocw.mit.edu 18.06SC Linear Algebra Fall 2011 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .