Algebra and Geometry | Part IA, 2002

Define what is meant by a vector space VV over the real numbers R\mathbb{R}. Define subspace, proper subspace, spanning set, basis, and dimension.

Define the sum U+WU+W and intersection UWU \cap W of two subspaces UU and WW of a vector space VV. Why is the intersection never empty?

Let V=R4V=\mathbb{R}^{4} and let U={xV:x1x2+x3x4=0}U=\left\{\mathbf{x} \in V: x_{1}-x_{2}+x_{3}-x_{4}=0\right\}, where x=(x1,x2,x3,x4)\mathbf{x}=\left(x_{1}, x_{2}, x_{3}, x_{4}\right), and let W={xV:x1x2x3+x4=0}W=\left\{\mathbf{x} \in V: x_{1}-x_{2}-x_{3}+x_{4}=0\right\}. Show that UWU \cap W has the orthogonal basis b1,b2\mathbf{b}_{1}, \mathbf{b}_{2} where b1=(1,1,0,0)\mathbf{b}_{1}=(1,1,0,0) and b2=(0,0,1,1)\mathbf{b}_{2}=(0,0,1,1). Extend this basis to find orthogonal bases of U,WU, W, and U+WU+W. Show that U+W=VU+W=V and hence verify that, in this case,

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

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