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Let $A = \begin{bmatrix}-1&4\\-2&3\end{bmatrix}, \vec{b}_1 = \begin{bmatrix}3\\2\end{bmatrix}, \vec{b}_2 = \begin{bmatrix}-1\\1\end{bmatrix},$ and $B = \{\vec{b}_1, \vec{b}_2\}.$ Find a matrix $A$ so that $[T(\vec{v})]_B = A[\vec{v}]_B$

I don't know the symbol for the B they are writing but I'm not sure if it matters. I don't even understand what this question is even asking me to solve. B equals some set of those two vectors? What is T(v) and A(v) and what does it mean $_B$ If anyone could even point me to a question that's smiliar but with different numbers im sure i can figure it out

Yusha
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    You already give a matrix $A$, but no linear transformation $T$. – Aweygan Jul 25 '16 at 18:30
  • So should I multiply A by the two given vectors or? – Yusha Jul 25 '16 at 18:31
  • Can you link me to a similar problem so I can see the method of solving this @Aweygan – Yusha Jul 25 '16 at 18:32
  • You need a matrix $A'$. The matrix $A$ is the standard matrix of $T$. $B$ stands for basis, and they're asking for the matrix of $T$ with respect to the basis $B$. Look in your textbook for the change-of-basis formula. – Ted Shifrin Jul 25 '16 at 18:32
  • Found this: https://www.khanacademy.org/math/linear-algebra/alternate-bases/change-of-basis/v/linear-algebra-change-of-basis-matrix – Yusha Jul 25 '16 at 18:34

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I'll just clarify the question for you.

$\mathcal B$ is a basis (thus why they named it $\mathcal B$ ;) ) and the notation $[v]_{\mathcal B}$ and $[T(v)]_{\mathcal B}$ denote the matrix representations (i.e. column matrix containing the coordinates) of the vectors $v$ and $T(v)$ respectively ($T$ is a linear transformation so $T(v)$ is a vector) wrt $\mathcal B$. They want you to express the linear transformation $T$ as a matrix whose domain and codomain are given wrt the basis $\mathcal B$.

Now, unfortunately this question is written poorly because $1)$ instead of writing a prescription for $T$ they give a matrix $A$ and $2)$ the $A$ in the first sentence is different than the one in the second. But we can infer what they mean from the context: the $A$ in the first sentence is the matrix representation for $T$ in the standard basis and you need to find another matrix (which they also call $A$ because whoever wrote this question is a terrible person) which represents $T$ but this time wrt $\mathcal B$.

  • So do I need to multiply vector $b_1, b_2$ to $A$ ? – Yusha Jul 25 '16 at 18:37
  • Do you know what is meant by "the matrix representation of a linear transformation (or vector)"? –  Jul 25 '16 at 18:37
  • I think I should first diagonlize $A$ – Yusha Jul 25 '16 at 18:59
  • If you want, but it's an unnecessary step. –  Jul 25 '16 at 19:00
  • What is meant by ""the matrix representation of a linear transformation (or vector)" – Yusha Jul 25 '16 at 19:00
  • oh yeah i know what that is – Yusha Jul 25 '16 at 19:03
  • This answer will hopefully explain what the matrix representation of a vector is. Then given a linear map $T$ and a vector $v$, the matrix representation of $T$ with respect to (wrt) a basis $\mathcal B$ is the matrix $M$ such that $M[v]{\mathcal B} = [T(v)]{\mathcal B}$. And this answer will hopefully help you understand change of basis of a matrix. –  Jul 25 '16 at 19:07
  • I got $P^{-1}AP = \begin{bmatrix}5&10\-10&5\end{bmatrix}$, @Bye_World, can you explain to me why in the book in the solution they are multiplying this matrix by $\frac{1}{5}$? – Yusha Jul 26 '16 at 19:04
  • @Yusha WolframAlpha also gets $\frac 15$ of that matrix, so I guess you made a mistake somewhere in your calculations. Try to go back through and see where you made a mistake. If you can't find it then ask a new question with the work you've done and someone will probably be able to spot the error for you. –  Jul 26 '16 at 20:00