% MATLAB Recitation Demo for Monday, September 15. % File: rdemo2b % % *** Linear combination of matrices *** % % MATLAB can help answer many questions concerning % matrices and vectors. % For example: % Express b as a linear combination of vectors (b = Ax), % Express A as a product of matrices (A = LU), or % Express A as a linear combination of other matrices. % % For the latter, we can "reshape" matrices into vectors % with m*n components -- and then check if vec(A) is a % linear combination of vec(S), vec(T), etc. >> diary rdemo2b >> A = [17 14; 11 8] A = 17 14 11 8 >> S = [1 2; 3 4] S = 1 2 3 4 >> T = [5 6; 7 8] T = 5 6 7 8 % The MATLAB command reshape(S, 4, 1) makes S into a matrix % whose shape has 4 rows and 1 column (i.e., a vector). % Important: Reshape takes entries from column 1, then % column 2, etc. >> v1 = reshape(S,4,1) v1 = 1 3 2 4 >> v2 = reshape(T,4,1) v2 = 5 7 6 8 >> b = reshape(A,4,1) b = 17 11 14 8 >> Z = [v1 v2 b] Z = 1 5 17 3 7 11 2 6 14 4 8 8 % Use elementary row operations to convert Z into % reduced row echelon form. % Recall that such elementary row operations yield % an equivalent linear system that can be solved by % backsubstitution or "inspection". >> Z = ref(Z) Z = 1 0 -8 0 1 5 0 0 0 0 0 0 % As usual, Z is now the augmented matrix for Rx = d. % % By inspection, we see that the vector d is an % obvious linear combination of the columns of R. % x = [-8; 5] % % Let's verify! >> x = [-8; 5] x = -8 5 >> -8 * S + 5 * T ans = 17 14 11 8 >> A A = 17 14 11 8