Linear mappings: Matrices and coordinate transformations
Transition to another coordinate system
Let \(L\) be the mapping from \(\mathbb{R}^2\) to \(\mathbb{R}^2\) defined by \[L(x,y)=(2x+y, x-y)\]
What is the matrix \([L]_e^e\) in terms of the unit vectors \(\vec{e}_1=\cv{1\\0}\) and \(\vec{e}_2=\cv{0\\1}\)?
What is the matrix \([L]_f^f\) in \(f\)-coordinates where \(\vec{f}_{\!1}=\matrix{-1 \\ 1 \\ }\) and \(\vec{f}_{\!2}=\matrix{1 \\ 0 \\ }\)?
What is the matrix \([L]_e^e\) in terms of the unit vectors \(\vec{e}_1=\cv{1\\0}\) and \(\vec{e}_2=\cv{0\\1}\)?
What is the matrix \([L]_f^f\) in \(f\)-coordinates where \(\vec{f}_{\!1}=\matrix{-1 \\ 1 \\ }\) and \(\vec{f}_{\!2}=\matrix{1 \\ 0 \\ }\)?
| \([L]_e^e={}\) |
| \([L]_f^f={}\) |
Unlock full access
