Simple questions to check comprehension of definition of Euclidean inner product and norm. Meant for formative use.
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", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "You are given the vectors $v= \\var{vector(a,b,g)}$ and $w = \\var{vector(c,d,f)}$ in $\\mathbb{R}^3$.
", "advice": "\\begin{align}
\\langle v,w\\rangle =v^T \\cdot w &= \\begin{pmatrix}\\var{a} &\\var{b} & \\var{g}\\end{pmatrix} \\begin{pmatrix}\\var{c}\\\\\\var{d}\\\\\\var{f}\\end{pmatrix} \\\\
&= \\simplify[timesdot]{{a}*{c}+{b}*{d}+{g}*{f}} \\\\
&= \\var{inner}
\\end{align}
Find the inner product $\\langle v,w\\rangle = v^T w = $ [[0]]
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", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "You are given the vector $v= \\var{vector(a,b,g)}$ in $\\mathbb{R}^3$.
", "advice": "\\begin{align}
\\|v\\|^2 =v^T \\cdot v &= \\begin{pmatrix}\\var{a} &\\var{b} & \\var{g}\\end{pmatrix} \\begin{pmatrix}\\var{a}\\\\\\var{b}\\\\\\var{g}\\end{pmatrix} \\\\
&= \\simplify[]{{a}^2+{b}^2+{g}^2} \\\\
&= \\var{squarenorm}
\\end{align}
So \\(\\|v\\|=\\sqrt{\\var{squarenorm}}\\).
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", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "You are given the vectors $v= \\var{v}$ and $w = \\var{w}$ in $\\mathbb{R}^4$.
", "advice": "\\begin{align}
\\langle v,w\\rangle =v^T \\cdot w &= \\begin{pmatrix}\\var{v[0]} &\\var{v[1]} & \\var{v[2]}& \\var{v[3]}\\end{pmatrix} \\var{w} \\\\
&= \\simplify[timesdot]{{v[0]}*{w[0]}+{v[1]}*{w[1]}+{v[2]}*{w[2]}+{v[3]}*{w[3]}} \\\\
&= \\var{inner}
\\end{align}
Find the inner product $\\langle v,w\\rangle = v^T w = $ [[0]]
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", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "You are given the vector $v= \\var{v}$ in $\\mathbb{R}^4$.
", "advice": "\\begin{align}
\\|v\\|^2 =v^T \\cdot v &= \\begin{pmatrix}\\var{v[0]} &\\var{v[1]} & \\var{v[2]}& \\var{v[3]}\\end{pmatrix} \\var{v} \\\\
&= \\simplify[]{{v[0]}^2+{v[1]}^2+{v[2]}^2+{v[3]}^2} \\\\
&= \\var{squarenorm}
\\end{align}
So \\(\\|v\\|=\\sqrt{\\var{squarenorm}}\\).
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