## Solution to Linear Algebra Done Wrong Exercise 1.3.7

Show that any linear transformation in $\mathbb C$ (treated as a complex vector space) is a multiplication by $\alpha\in\mathbb C$. Solution: Let $T$ be a linear transformation from $\mathbb C$…

Show that any linear transformation in $\mathbb C$ (treated as a complex vector space) is a multiplication by $\alpha\in\mathbb C$. Solution: Let $T$ be a linear transformation from $\mathbb C$…

Let $A$ be a linear transformation. If ${\bf z}$ is the center of the straight interval $[{\bf x}, {\bf y}]$, show that $A{\bf z}$ is the center of the interval…

Find $3\times 3$ matrices representing the transformations of $\mathbb R^3$ which:a) project every vector onto $x\text{-}y$ plane;b) reflect every vector through $x\text{-}y$ plane;c) rotate the $x\text{-}y$ plane through $30^\circ$, leaving…

For each linear transformation below find its matrix a) $T:\mathbb R^2\to \mathbb R^3$ defined by $$T(x,y)^T=(x+2y,2x-5y,7y)^T;$$ Solution: We have $$T\begin{pmatrix}x\\ y\end{pmatrix}=x\begin{pmatrix} 1\\ 2\\ 0\end{pmatrix}+y\begin{pmatrix} 2\\ -5\\ 7\end{pmatrix},$$ hence the matrix…

Let a linear transformation in $\mathbb R^2$ be the reflection in the line $x_1=x_2$. Find its matrix. Solution: Let us think $x_1=x$ and $x_2=y$. In the $(x,y)$-coordinate plane, the graph…