MAT223H1 Study Guide - Midterm Guide: 3Cx Phone System, Elementary Matrix, Linear Combination
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March 1, 2007: beltran, g. naot, r. roeder, s. uppal. [4] 1(a) if b / im(a), then ax = b has no solution. true false. If ax = b has a solution then b is a linear combination of the columns of a and so is in im(a). [4] 1(b) if a is an n n matrix and null(a) = {0}, then a is invertible. true false null(a) = {x rn | ax = 0}. So if null(a) = {0} then ax = 0 only for x = 0 which implies a is invertible. For each of the statements below, decide if it is true or false. Indicate your answer by shading in the box corresponding to your choice. Justify your answer by providing an appropriate proof or counter example. [4] 2(a) every plane in r3 is a subspace of r3 of dimension 2. true false. This is only true if the plane contains the origin.
