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July 9th, 2018, 09:32 AM   #1
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Please help with this problem

Considering the two basis of R. The 1st , B is the canonical base i,j defined by i={1 0} (i is column) and j={0 1} (j is column). The 2nd, B´ is associated to matrix [P]=[1 -1; 1 1] (1 -1) ***(1 -1) is first column and (1 1) is second column. A vector V is expressed as V= {1 2} ***(1 2) is column*** in B. What is the expression of the vector V in basis B´?
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July 11th, 2018, 03:31 AM   #2
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Surely you mean $\displaystyle R^2$ rather than R? And you have two bases, not two basis vectors? Vector v can be written as $\displaystyle v= \begin{pmatrix}1 \\ 0 \end{pmatrix}+ 2\begin{pmatrix} 0 \\ 1 \end{pmatrix}= \begin{pmatrix}1 \\ 2\end{pmatrix}$.

We need to find numbers a and b such that $\displaystyle v= \begin{pmatrix}1 \\ 2 \end{pmatrix}=$$\displaystyle a\begin{pmatrix}1 \\ -1 \end{pmatrix}$$\displaystyle + b\begin{pmatrix}1 \\ 1\end{pmatrix}=$$\displaystyle \begin{pmatrix} a+ b \\ -a+ b \end{pmatrix}$

That is, we need to solve the equations a+ b= 1 and -a+ b= 2.
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