1.II.12H

Linear Algebra | Part IB, 2004

Let UU and WW be subspaces of the finite-dimensional vector space VV. Prove that both the sum U+WU+W and the intersection UWU \cap W are subspaces of VV. Prove further that

dimU+dimW=dim(U+W)+dim(UW)\operatorname{dim} U+\operatorname{dim} W=\operatorname{dim}(U+W)+\operatorname{dim}(U \cap W)

Let U,WU, W be the kernels of the maps A,B:R4R2A, B: \mathbb{R}^{4} \rightarrow \mathbb{R}^{2} given by the matrices AA and BB respectively, where

A=(12131124),B=(11200124)A=\left(\begin{array}{rrrr} 1 & 2 & -1 & -3 \\ -1 & 1 & 2 & -4 \end{array}\right), \quad B=\left(\begin{array}{rrrr} 1 & -1 & 2 & 0 \\ 0 & 1 & 2 & -4 \end{array}\right)

Find a basis for the intersection UWU \cap W, and extend this first to a basis of UU, and then to a basis of U+WU+W.

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