Question

help needed with linear algebra!

Answer 1

to show that it is a linear operator, we have to show that
\[g(r\vec{y})=rg(\vec{y})\]

Answer 2

or that \[g(\mathbf{x}+\mathbf{y})=g(\mathbf{x})+g(\mathbf{y})
\]

Answer 3

we have, by definition,\[
g(\mathbf{y})=\mathbf{P}\cdot \mathbf{x}\\
g(\mathbf{y})=\left[\begin{matrix}a&0&0\\0&b&0\\0&0&c
\end{matrix}\right]\cdot \mathbf{x}\\
\;\\
g(\mathbf{x}+\mathbf{y})=\mathbf{P}\cdot(\mathbf{x}+\mathbf{y})\\
\text{we know that matrix multiplication is distributive}\\
g(\mathbf{x}+\mathbf{y})=P\cdot \mathbf{x}+P\cdot \mathbf{y}\\
g(\mathbf{x}+\mathbf{y})=g(\mathbf{x})+g(\mathbf{y})
\]
QED

Answer 4

@ lynn this time, no way to get just the answer, right ? It's a "show" problem. Good luck

Answer 5

\[\huge\color{red}{Amazing~Art~in~Avatar}\]\[\huge\color{purple}{Got~It}\]