Let T : Rn →Rm be a linear transformation. (a) If {v1,v2,...,vk} is a linearly dependent...

Let T : Rn →Rm be a linear transformation.

(a) If {v1,v2,...,vk} is a linearly dependent subset of Rn, prove that {T(v1),T(v2),...,T(vk)} is a linearly dependent subset of Rm.

(b) Suppose the kernel of T is {0}. (Recall that the kernel of a linear transformation T : Rn → Rm is the set of all x ∈ Rn such that T(x) = 0). If {w1,w2,...,wp} is a linearly independent subset of Rn, then show that {T(w1),T(w2),...,T(wp)} is a linearly independent subset of Rm.

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Colby Messinger answered 1 year ago

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