Please write out full proof and theorems used. Problem 3. Let U be a subspace of

Question

Please write out full proof and theorems used.

Problem 3. Let U be a subspace of a vector space V and B denote a basis of V. Prove that any linear transformation from U to V can be extended to a linear transformation from V to V. In other words, show that for any linear transformation S: U right arrow V, there exists a linear transformation T: V right arrow V such that Tu = Su for all u in U. (Note: any linear independent set {g1, ... ,gm} in V can be extended to a basis (g1,?,gm, gm + 1, ?, gn) of V. You may use this fact here, without proof, )

Explanation / Answer

We know that, ant linear transformation maps a linearly independent set of V to another linearly independent set and the linear transformation maps a basis of V to another basis.

Also, the set of all linear combinations of the vectors of U forms a subspace of V and this is the smallest subspace which is called as linear span.

A subset of linearly independent set of vectors are linearly independent.

Combining all the results stated above, we get our theorem proved.

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