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Let \\( V \\) be an \\( n \\)-dimensional real inner product space, and let \\( \\vec{v}_0\\in V \\) be a non-zero vector.<\/p>\n\n

Consider the linear map \\( L\\colon V\\to\\mathbb{R}\\) defined by \\( L(\\vec{x}) = \\langle \\vec{x}, \\vec{v}_0 \\rangle \\).<\/p>\n\n

What is \\( \\text{rank}(L) \\)?<\/p>\n"},"library":"H5P.AdvancedText 1.1","metadata":{"contentType":"Text","license":"U","title":"Untitled Text"},"subContentId":"9e1388cb-0308-4644-b9c0-39870bd7e6af"},"useSeparator":"auto"},{"content":{"params":{"panels":[{"content":{"params":{"text":"

Answer:<\/strong> \\( \\text{rank}(L)=1 \\).<\/p>\n\n

Reasoning:<\/strong> We claim that \\( L \\) is surjective, and so \\( \\text{rank}(L) = \\text{dim}(\\text{range}(L)) = \\text{dim}(\\mathbb{R}) = 1 \\). To see why \\( L \\) is surjective, first notice that \\( L \\) is not the zero map since \\( L(\\vec{v}_0)= \\langle \\vec{v}_0, \\vec{v}_0 \\rangle \\neq\\vec{v}_0 \\). Thus \\( \\text{range}(L) \\) is not the zero subspace of the codomain \\( \\mathbb{R}\\). The only other option, since \\( \\mathbb{R}\\) is one-dimensional, is for \\( \\text{range}(L) \\) to be equal to the whole of \\( \\mathbb{R}\\).<\/p>\n\n

Alternatively:<\/strong> We note that the kernel of \\( L \\) consists of all vectors orthogonal to \\( \\vec{v}_0 \\). From this, we can deduce that \\[ \\ker(L) = (\\mathrm{Span}\\{\\vec{v}_0\\})^\\perp. \\] Consequently, \\[ \\text{dim}(\\ker(L)) = \\text{dim}((\\mathrm{Span}\\{\\vec{v}_0\\})^\\perp) = \\text{dim}(V) - \\text{dim}(\\mathrm{Span}\\{\\vec{v}_0\\}) = n-1, \\] where we've used the fact that \\( \\vec{v}_0\\neq\\vec{v}_0 \\) to conclude that \\( \\text{dim}(\\mathrm{Span}\\{\\vec{v}_0\\})=1 \\). Now, the rank--nullity theorem yields \\[ \\text{rank}(L) = n - \\text{dim}(\\ker(L)) = n-(n-1) = 1. \\]<\/p>\n"},"library":"H5P.AdvancedText 1.1","subContentId":"9ea5fa66-f247-42d8-96ea-c9dd0982e9a9","metadata":{"contentType":"Text","license":"U","title":"Untitled Text"}},"title":"Click to see the solution"}],"hTag":"h4"},"library":"H5P.Accordion 1.0","metadata":{"contentType":"Accordion","license":"U","title":"4.2f-Inner-Products-and-Norms-solution"},"subContentId":"72ba224d-1d69-4db3-aa11-dd81c539ed76"},"useSeparator":"auto"}]}