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Math Help - linear mapping

  1. #1
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    Wink linear mapping

    Hi,
    I really need your help with this:
    Suppose that V and W are finite-dimensional and that U is a subspace of V. Prove that there exists a T linear map from V to W such that null(T)=U if and only if dim(U)>=dim(V)-dim(W).
    Thank you in advance!
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  2. #2
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    I will prove the forward direction, you try proving the other direction.

    Say that T:V\to W is a linear transformation. Then by rank-nullity theorem we know \text{rank}(T) + \text{nullity}(T) = \text{dim}(V). But we are told that \text{null}(T) = U thus \text{nullity}(T) = \text{dim}(U). Putting this together we have that \text{dim}(U) = \text{dim}(V) - \text{rank}(T). But T[ U ] \subseteq W so \text{rank}(T) = \text{dim}(T[ U ]) \leq \text{dim}(W) this means -\text{rank}(T) \geq - \text{dim}(W). Finally, \text{dim}(U) \geq \text{dim}(V) - \text{dim}(W).
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  3. #3
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    Quote Originally Posted by bamby View Post
    Hi,
    I really need your help with this:
    Suppose that V and W are finite-dimensional and that U is a subspace of V. Prove that there exists a T – linear map from V to W such that null(T)=U if and only if dim(U)>=dim(V)-dim(W).
    Thank you in advance!
    (\Longrightarrow) trivial by rank-nullity theorem.

    (\Longleftarrow) suppose \dim W \geq \dim V - \dim U. let B_1=\{u_1, \cdots , u_m \} be a basis for U and extend B_1 to a basis for V: \ \ B=\{u_1, \cdots , u_m, v_1, \cdots , v_n \}.

    so we have that \dim W \geq n. thus there exists a set of n linearly independent elements of W, say w_1, \cdots , w_n. now define the map T:V \longrightarrow W by:

    T(a_1u_1 + \cdots a_m u_m + b_1v_1 + \cdots + b_n v_n)=b_1w_1 + \cdots + b_nw_n. it's obvious that T is a linear map and \ker T=U. \ \ \ \square
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