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Unformatted text preview: Selected answers to assignment 5, 2.2–2.3 2.2 5. (c) (1 0 0 1) (d) (1 2 4) (f) 3- 6 1 (g) ( a ) 8. If x is a linear combination of basis vectors with coefficients a i and y with coefficients b i , then T ( cx + y ) = ca 1 + b 1 . . . ca n + b n = c a 1 . . . a n + b 1 . . . b n = cT ( x ) + T ( y ) . 11. Let α be a basis for W and extend it to a basis β for all of V , ordered so that the vectors from α are listed first. Since T ( W ) ⊆ W , all elements of α will be mapped by T to linear combinations of elements of α . That is, their image coordinate vectors will have zeros in the ( k + 1) st through n th places. Since the images of the basis vectors form the columns of [ T ] β , the first k columns will be zero from row k + 1 down, and [ T ] β has the required form. 13. We want to show that if aT + bU = T , we must have a = b = 0. We know that if one of a or b is nonzero, the other must also be, because neither T nor U is T and hence neither is a multiple of T . If ( aT + bU )( x ) = , then aT...
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