Let (V; <;>) be an inner product space and v, w be two non- zero vectors in V .

Question

Let (V; <;>) be an inner product space and v, w be two non- zero vectors in V . Define T : V -> V by T(x) =< x, v > w. (a) Show that T is linear. (b) Find the image of T. (c) If dim(V ) = n, what is the dimension of ker(T)?

Explanation / Answer

(a)For linear, just show L(ax+by)=aL(x)+bL(y) L(ax+by)=w=(a+b)w=aw + bw=aL(x)+bL(y) (b)Since inner product space is VxV----> F Therefore, Image of T=cw, i.e. subspace spanned by w vector which is subset of V (c)Since inner product space is VxV----> F therefore, T(x)=w is multiple of w which can be written in form of n basis vector of V => Therefore, dim(range of T)=n By nullity theorem ker(T)=dimV-rank(T)=0

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