Math Challenges

Submissions for Problem #87

Problem #87

\[ \displaylines{\text{Given a basis }B=\left\lbrace\left(1,2\right),\left(3,4\right)\right\rbrace\text{ and }\mathbf{v}=\left(1,0\right)\\ 1)\text{ Determine }\left\lbrack\mathbf{v}\right\rbrack_{B}\\ 2)\text{ Find }P_{B}\text{ (The change of coordinates matrix from B-coordinates to standard basis)}} \]

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zapwai

Solution: \( \displaylines{\text{Express }\mathbf{v}\text{ in the basis }B.\\ \begin{pmatrix}1\\ 0\end{pmatrix}=a\begin{pmatrix}1\\ 2\end{pmatrix}+b\begin{pmatrix}3\\ 4\end{pmatrix}\implies a+3b=1,2a+4b=0\implies a=-2,b=1\\ \left\lbrack\mathbf{v}\right\rbrack_{B}=\left(-2,1\right)\\ P_{B}=\begin{pmatrix}1 & 3\\ 2 & 4\end{pmatrix}} \)