MAT223H1 Study Guide - Midterm Guide: Elementary Matrix, Row And Column Spaces, Euclidean Vector
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March 21, 2014: garcia-raboso, c. kent, f. murnaghan, k. tyros, s. uppal. 1 of 14: suppose you are given that the matrix a = echelon form. [9] (b) find a basis for each of the following three subspaces: (i) row(a), (ii) col(a), and (iii) null(a). Solution: (a) rank(a) = # linearly independent rows = # pivots in r = 3 (b) (i) since dim(row(a)) = rank(a) = 3, all the rows of a (or r) form a basis for the row. 3 of a corresponding to each pivot of r. hence: basiscol(a) = Let x6 = s, x5 = t, x3 = u. Then: x1 = 5u 22t s x2 = u + 5t x4 = 2t. , and hence: (cid:34) x1 (cid:35) x2 x3 x4 x5 x6. 3: consider the subset w = x2 x3. [5] (a) show that w is a subspace of r3 (cid:111) x2. We show that w is a subspace by de nition.
