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

Determine the dimension of the subspace \\( W \\) of the vector space \\( V \\) for the following definitions.<\/p>\n"},"library":"H5P.AdvancedText 1.1","metadata":{"contentType":"Text","license":"U","title":"Untitled Text"},"subContentId":"6bbd0afc-193c-478e-a11e-912413f5b583"},"useSeparator":"auto"},{"content":{"params":{"text":"

a.<\/strong> \\( W=\\{ A\\in M_{2\\times 2}(\\mathbb{F}) \\colon A^T=A\\} \\), \\( V=M_{2\\times 2}(\\mathbb{F}) \\).<\/p>\n"},"library":"H5P.AdvancedText 1.1","metadata":{"contentType":"Text","license":"U","title":"Untitled Text"},"subContentId":"c261a641-73dc-41bb-99f1-a5605f6a7e29"},"useSeparator":"auto"},{"content":{"params":{"panels":[{"content":{"params":{"text":"

Answer:<\/strong> 3<\/p>\n\n

Reasoning:<\/strong> A basis for \\( W \\) is given by \\( \\left\\{\\begin{bmatrix} 1&0\\\\0&0 \\end{bmatrix},\\begin{bmatrix} 0&1\\\\1&0 \\end{bmatrix}, \\begin{bmatrix} 0&0\\\\0&1 \\end{bmatrix}\\right\\} \\).<\/p>\n"},"library":"H5P.AdvancedText 1.1","subContentId":"7fb1eadc-368c-45b4-858c-c666bc28f5e2","metadata":{"contentType":"Text","license":"U","title":"Untitled Text"}},"title":"Click to see the solution for part a"}],"hTag":"h4"},"library":"H5P.Accordion 1.0","metadata":{"contentType":"Accordion","license":"U","title":"1.4a-Basis-and-Dimension-part-a"},"subContentId":"9a37ee9b-b1c4-42d5-b27b-8a210e5d76b8"},"useSeparator":"disabled"},{"content":{"params":{"text":"

b.<\/strong> \\( W=\\{ A \\in M_{4\\times 4}(\\mathbb{F}) \\colon A \\text{ is a diagonal matrix}\\} \\), \\( V=M_{4\\times4}(\\mathbb{F}) \\).<\/p>\n"},"library":"H5P.AdvancedText 1.1","metadata":{"contentType":"Text","license":"U","title":"Untitled Text"},"subContentId":"27e70fbf-ee20-430a-85b2-d78f16b34f7f"},"useSeparator":"auto"},{"content":{"params":{"panels":[{"content":{"params":{"text":"

Answer:<\/strong> 4<\/p>\n\n

Reasoning:<\/strong> A basis for \\( W \\) is given by \\( \\left\\{E_{11}, E_{22}, E_{33}, E_{44}\\right\\} \\), where \\( E_{ii} \\) is the matrix whose \\( (i,i) \\) entry is \\( 1 \\) and whose other entries are all \\( 0 \\).<\/p>\n"},"library":"H5P.AdvancedText 1.1","subContentId":"94a06de4-ba9b-4105-9c07-8d75e2e06e0c","metadata":{"contentType":"Text","license":"U","title":"Untitled Text"}},"title":"Click to see the solution for part b"}],"hTag":"h4"},"library":"H5P.Accordion 1.0","metadata":{"contentType":"Accordion","license":"U","title":"1.4a-Basis-and-Dimension-part-b"},"subContentId":"6988b4c6-e2df-4594-a789-59c987ad2642"},"useSeparator":"disabled"},{"content":{"params":{"text":"

c.<\/strong> \\( W=\\{p(x)\\in \\mathcal{P}_2(\\mathbb{F}) \\colon p'(1)=p(1)=0\\} \\), \\( V=\\mathcal{P}_2(\\mathbb{F}) \\).<\/p>\n\n

Note:<\/strong> \\( \\mathcal{P}_n(\\mathbb{F}) \\) is the vector space of polynomials in the indeterminate \\( x \\) of degree \\( \\leq n \\) with coefficients in \\( \\mathbb{F} \\). <\/p>\n"},"library":"H5P.AdvancedText 1.1","metadata":{"contentType":"Text","license":"U","title":"Untitled Text"},"subContentId":"5c5ad619-133b-4ad4-973e-cb104b5a8d02"},"useSeparator":"auto"},{"content":{"params":{"panels":[{"content":{"params":{"text":"

Answer:<\/strong> 1<\/p>\n\n

Reasoning:<\/strong> A basis for \\( W \\) is given by \\( \\{x^2-2x+1\\} \\).<\/p>\n"},"library":"H5P.AdvancedText 1.1","subContentId":"58e1a417-d3a7-4985-a2e8-146bd180be0e","metadata":{"contentType":"Text","license":"U","title":"Untitled Text"}},"title":"Click to see the solution for part c"}],"hTag":"h4"},"library":"H5P.Accordion 1.0","metadata":{"contentType":"Accordion","license":"U","title":"1.4a-Basis-and-Dimension-part-c"},"subContentId":"77f10f76-a8b7-4189-bdcd-22404f6496a0"},"useSeparator":"disabled"},{"content":{"params":{"text":"

d.<\/strong> \\( W=\\mathrm{Span}\\{x-x^2,x-x^3,x^2-x^3\\} \\), \\( V=\\mathcal{P}_3(\\mathbb{F}) \\).<\/p>\n\n

Note:<\/strong> \\( \\mathcal{P}_n(\\mathbb{F}) \\) is the vector space of polynomials in the indeterminate \\( x \\) of degree \\( \\leq n \\) with coefficients in \\( \\mathbb{F} \\). <\/p>\n"},"library":"H5P.AdvancedText 1.1","metadata":{"contentType":"Text","license":"U","title":"Untitled Text"},"subContentId":"75457b8c-503c-4609-8f53-31fefa12a713"},"useSeparator":"auto"},{"content":{"params":{"panels":[{"content":{"params":{"text":"

Answer:<\/strong> 2<\/p>\n\n

Reasoning:<\/strong> A basis for \\( W \\) is given by \\( \\{x-x^2, \\ x^2 - x^3\\} \\).<\/p>\n"},"library":"H5P.AdvancedText 1.1","subContentId":"b35d486a-51fc-467e-98b3-5730e9b2c2a7","metadata":{"contentType":"Text","license":"U","title":"Untitled Text"}},"title":"Click to see the solution for part d"}],"hTag":"h4"},"library":"H5P.Accordion 1.0","metadata":{"contentType":"Accordion","license":"U","title":"1.4a-Basis-and-Dimension-part-d"},"subContentId":"4f69eac2-e4ac-4856-903f-ae68783bbb26"},"useSeparator":"disabled"}]}