Let V and W be finite dimensional vector spaces. Let T : V → W be a linear transformation of V into W. (a) Let H be a subspace of V. Let T(H) be the set of all images of the vectors in H under the transformation T. Show that T(H)is a subspace of W and dim T(H)< dim H (b) Now suppose that T is one-to-one. Show that dim T(H)= dim H.

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Q: If W₁ and W₂ are subspace of a vector space V(F), then show that W₁ + W₂ is also a subspace of V(F).

A: W1 and W2 are the subspace of a vector space BF We have to show W1+W2 is also a subspace of BF

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A: Given that W1 and W2 are two subspace of a vector space V over a field F.

Q: Let W be a subspace of a vector space V. Show that if x is in both W and W+, then x = 0.

A: Solution is given below

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A: Given: A:V→W is a linear transformation and dim(V)=n. To show: dim Im(A)+dim Ker(A)=n.

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A: This statement is False.  By using property of subspace we can prove this statement is False.

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A: Answer

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A: we have to  prove that T is a linear transformation .Then have to determine whether T is a

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A: Follow the steps.