a) Let [tex]\mathbf{P}[/tex] be the [tex]2\times 2[/tex] matrix that projects vectors onto [tex]\mathbf{u} = \begin{pmatrix}2\\ -1 \end{pmatrix}[/tex]. That is, [tex]\mathbf{P} \bold{v} = \operatorname{proj}_{\bold{u}} (\bold{v}) = \text{Projection of $\mathbf{v}$ onto $\mathbf{u}$}.[/tex] Use the picture below (bottom attachment) to find [tex]\mathbf{P} \begin{pmatrix}2 \\ -1 \end{pmatrix} \text{ and } \mathbf{P} \begin{pmatrix}1 \\ 2 \end{pmatrix}[/tex] using the geometric meaning of the matrix. Enter the answers as columns in the order above.
b) Let [tex]\mathbf{P}[/tex] be the [tex]2\times 2[/tex] matrix that projects vectors onto [tex]\mathbf{u} = \begin{pmatrix}2\\ -1 \end{pmatrix}[/tex]. That is, [tex]\mathbf{P} \bold{v} = \operatorname{proj}_{\bold{u}} (\bold{v}) = \text{Projection of $\mathbf{v}$ onto $\mathbf{u}$}.[/tex] Use your answers from part (a) to figure out [tex]\mathbf{P}[/tex].
c) Use the matrix [tex]\mathbf{P}[/tex] you found in part (b) to figure out [tex]\mathbf{w}_1[/tex] and [tex]\mathbf{w}_2[/tex] in the diagram below (top attachment).