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# Precalc Scalars and Matrices

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Every vector v can be expressed uniquely in the form a + b where a is a scalar multiple of $$\begin{pmatrix} 2 \\ -1 \end{pmatrix},$$ and b is a scalar multiple of $$\begin{pmatrix} 3 \\ 1 \end{pmatrix}.$$Find the matrix P such that Pv = a for all vectors v.

I tried to combine the two scalars, multiplied by a and b, and I got $$\begin{bmatrix} 2a + 3b \\ -a + b \end{bmatrix}$$. I don't know how to get the matrix from this. Is it just $$\begin{bmatrix} 2 & 3 \\ -1 & 1 \end{bmatrix}$$?

Mar 14, 2022

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Actually

$$\mathbf{P} = \begin{pmatrix} 2 & -1 \\ 3 & 1 \end{pmatrix}$$

Mar 15, 2022
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You have   $$\displaystyle \textbf{v}=\left[\begin{array}{c} 2a+3b\\ -a+b \end{array}\right]$$ and require that  $$\displaystyle P\, \textbf{v}=\left[\begin{array}{c} 2a \\ -a \end{array} \right]\, .$$

So, let  $$\displaystyle P=\left[\begin{array}{cc} r & s \\ t & u \end{array}\right]$$ say, multiply out the rhs and equate components across the equation to find r, s, t and u.

You should find that $$\displaystyle P = \left[ \begin{array}{cc} 2/5 & -6/5 \\ -1/5 & 3/5 \end{array} \right]\, .$$

Mar 16, 2022