Unformatted text preview: → W be a linear transformation where V and W are finite dimensional. a) If dim(V) = 5, dim(W) = 3, and dim(ker(T))= 2, then T is onto. b) If ker(T) = V, then W = { 0 }. 2) In each case, (i) find a basis of ker(T), and (ii) find a basis of range(T) a) T: M 22 → R; d a ; d c b a T + = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ b) T: P 2 → R 2 ; T(a + bx + cx 2 ) = (a, b). 3) Let T: M nn → R denote the trace map: T(A) = tr(A) for all A in M nn . Show that dim(ker(T)) = n 2 – 1. (hint: dimension theorem) 4) Given { } n v ..., in a vector space V, define T: R n → V by v v , , 2 1 ( ) n n n v r v r v r r r r T + + + = ... ..., , , 2 2 1 1 2 1 . Show that T is linear, and that: a) T is onetoone if and only if { } n v v v ..., , , 2 1 is independent. b) T is onto if and only if V = sp ( ) n v v v ..., , , 2 1 ....
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 Spring '10
 X.Jiang
 Linear Algebra, Algebra, Fraleigh

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