{"content":[{"content":{"params":{"text":"

Consider the subspace \\( W=\\{\\vec{x}\\in\\mathbb{R}^3\\colon x+y+z=0\\} \\) of \\( V=\\mathbb{R}^3 \\).<\/p>\n\n

Find the dimension of the orthogonal complement \\( W^\\perp \\) of \\( W \\) in \\( V \\) with respect to the dot product.<\/p>\n"},"library":"H5P.AdvancedText 1.1","metadata":{"contentType":"Text","license":"U","title":"Untitled Text"},"subContentId":"0f407002-55b5-481e-846a-88ba67cee6f7"},"useSeparator":"auto"},{"content":{"params":{"panels":[{"content":{"params":{"text":"

Answer: <\/strong>\\( \\text{dim}(W^\\perp)=1 \\).<\/p>\n\n

Reasoning: <\/strong>We can use the fact that \\( \\text{dim}(W)+\\text{dim}(W^\\perp)=\\text{dim}(V) \\). Here \\( \\text{dim}(V)=3 \\) and \\( \\text{dim}(W)=2 \\) since \\( W \\) is a plane in \\( \\mathbb{R}^3 \\). Thus, \\( \\text{dim}(W^\\perp)=\\text{dim}(V)-\\text{dim}(W)=3-2=1 \\).<\/p>\n\n

Alternatively, notice that \\( W^\\perp \\) is spanned by a normal vector to the plane \\( W \\). So \\( W^\\perp \\) is one-dimensional.<\/p>\n"},"library":"H5P.AdvancedText 1.1","subContentId":"eb8affef-022c-43df-8b38-4ae2d3986683","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":"4.2b-Orthogonality-solutioin"},"subContentId":"cf23e53e-6600-4146-b504-6a9c7ee41232"},"useSeparator":"auto"}]}