Linear and Multilinear Algebra
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Problems 4
- Do v1,v2,…,vnv_1,v_2,\dots,v_nv1,v2,…,vn span all of the vector space VVV? (Source: Robi Bhattacharjee)
- What is the maximum possible dimension of the span of 333 vectors in a 222-dimensional vector space?
- Let V,WV,WV,W be vector spaces over a field KKK. When is the natural map V∗⊗KW→HomK(V,W)V^\ast \otimes_K W \to \operatorname{Hom}_K(V,W)V∗⊗KW→HomK(V,W) an isomorphism? Give t…
- If x⃗,y⃗∈Rn\vec{x}, \vec{y} \in \mathbb{R}^nx,y∈Rn are such that Q(x⃗,x⃗)>0Q( \vec{x}, \vec{x}) > 0Q(x,x)>0 and Q(y⃗,y⃗)>0Q( \vec{y}, \vec{y}) > 0Q(y,y)>0, is it possible tha…