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jimmypoopins
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Homework Statement
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]
Homework 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.
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.