Problem 4. Let U and V be vector spaces, and let T : U - V be a linear transformation. Let (u1, ., Um) be a list of vectors in U such that (T(ul), .,...
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Let U and V be vector spaces, and let T : U → V be a linear transformation. Let (u1,...,um) be a list of vectors in U such that (T(u1),...,T(um)) is an ordered basis of imT. Also let (v1,...,vn) be an ordered basis of kerT.

1. (a) Provethatui ̸=vj forall1≤i≤mand1≤j≤n.
2. (b) Prove that (u1, . . . , um, v1, . . . , vn) is a linearly independent list of vectors in U.
3. [Hint: Apply T to a relation on them.]
4. (c) Prove that (u1,...,um,v1,...,vn) spans U.
5. (d) Prove the following version of the Rank-Nullity Theorem: If U and V are vector spaces where U is finite-dimensional, then for any linear transformation T : U → V , we have dim(U ) = dim(ker T ) + dim(im T ).

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Problem 4. Let U and V be vector spaces, and let T : U - V be a linear transformation. Let (u1, . . ., Um) be a list of vectors in U such that (T(ul), ..., T(um) ) is an ordered basis of im T. Also let (v1, . . ., Un) be an ordered basis of ker T. (a) Prove that ui # v; for all 1 &lt; i &lt; m and 1 &lt; j &lt; n. (b) Prove that (U1, . . ., Um, V1, . .., Un) is a linearly independent list of vectors in U. [HINT: Apply T to a relation on them. (c) Prove that (ul, . . ., Um, Vl, .. ., Un) spans U. (d) Prove the following version of the Rank-Nullity Theorem: If U and V are vector spaces where U is finite-dimensional, then for any linear transformation T : U - V, we have dim(U) = dim(ker T) + dim(im T).

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Subject: Linear Algebra, Math

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