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Please give answer with the details. Thanks a lot! Let T: V-W be a linear transformation between vector spaces V and W...

Let T: V-W be a linear transformation between vector spaces V and W (1) Prove that if T is injective (one-to-one) and {vi,..

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Let T: V-W be a linear transformation between vector spaces V and W (1) Prove that if T is injective (one-to-one) and {vi,.. ., vm) is a linearly independent subset of V the n {T(6),…,T(ền)} is a linearly independent subset of W (2) Prove that if the image of any linearly independent subset of V is linearly independent then Tis injective. (3) Suppose that {b1,... bkbk+1,. . . ,b,) is a basis of V such that ,bk^ is a basis of ker(T). Prove that Tbk+1), , T(%)} is a basis of im(T) (4) Let v, ..., Vk) be a basis of V. Prove that T is surjective (onto) if and only if the vectors T(v) ,T(%) span W
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Please give answer with the details. Thanks a lot! Let T: V-W be a linear transformation between vector spaces V and W...
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