(Solved): Linear Maps and Images Map Images Given the map \( f: \mathcal{P}_{2} \mapsto \mathcal{M}_{2 \time ...

Linear Maps and Images Map Images Given the map \( f: \mathcal{P}_{2} \mapsto \mathcal{M}_{2 \times 2} \) given by \[ a+b x+c x^{2} \mapsto\left(\begin{array}{ll} a+b & 2 a+b \\ a+c & a+2 c \end{array}\right) \] what is the image of the vector \( 5+3 x+2 x^{2} \) ? \[ f\left(5+3 x+2 x^{2}\right)=( \] Given a linear map \( f \) which acts so that \[ f\left(2 \vec{v}_{1}+\vec{v}_{2}\right)=\left(\begin{array}{l} 10 \\ 16 \end{array}\right) \] and so that \[ f\left(\vec{v}_{1}\right)=\left(\begin{array}{l} 2 \\ 1 \end{array}\right) \] then \[ f\left(\vec{v}_{2}\right)=\left(\begin{array}{l} x \\ y \end{array}\right) \] with \[ x= \] \[ y= \] Is this map one-to-one? Consider the linear map \[ f:\left(\begin{array}{lll} x & y & z \end{array}\right) \longrightarrow\left(\begin{array}{c} y \\ x \\ z \\ x \end{array}\right) \] Is this a map one-to-one?