(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose that V and W are finite dimensional and that U is a subspace of V. Prove that there exists [tex]T \in L(V,W)[/tex] such that null T = U if and only if [tex]dim U \geq dim V - dim W.[/tex]

2. Relevant equations

thm: If [tex]T \in L(V,W)[/tex], then range T is a subspace of W.

thm: If V is a finite dimensional vector space and [tex]T \in L(V,W)[/tex] then range T is a finite-dimensional subspace of W and dim V = dim null T + dim range T.

3. The attempt at a solution

forward direction: by thm, range T is a subspace of W implies that

[tex]dim range T \leq dim range W[/tex].

by thm, dim V = dim null T + dim range T

dim V = dim U + dim range T (since U = null T)

dim V - dim range T = dim U

[tex]dim V - dim W \leq dim U[/tex] since [tex]dim range T \leq dim range W[/tex].

i think the forward direction is good. comments?

backward direction:

we have [tex]dim V - dim W \leq dim U[/tex]. Let [tex](u_{1},...,u_{n})[/tex] be a basis for U. extend this to a basis for V: [tex](u_{1},...,u_{n},u_{n+1},...u_{m})[/tex]. then dim U = n, and dim V = m. Then any [tex]v \in V[/tex] can be written as [tex]a_{1}u_{1}+...+a_{m}u_{m}[/tex].

I think i'm in the right direction but i'm confused as to what to do. since we have dim V - dim W is less than dim U, i want to say that dim W is greater than or equal to m, but i don't know how to define T so that null T = U. If i make all of the T(u_i} in the basis 0, then null T = U, but how does that relate to the relation of [tex]dim V - dim W \leq dim U[/tex]?

thanks.

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