Geometry of Linear Maps

Use the $H=PBQ$ decomposition to find the combination of dilations, flips, skews, and projections that produces the map $h\colon\mathbb{R}^{3}\to\mathbb{R}^{3}$ represented with respect to the standard bases by this matrix.

What combination of dilations, flips, skews, and projections produces a rotation counterclockwise by $2\pi/3$ radians?

Show that any linear transformation of $\mathbb{R}^{1}$ is a map $h_{k}$ that multiplies by a scalar $x\mapsto kx$.

Show that linear maps preserve the linear structures of a space.

• Show that for any linear map from $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$, the image of any line is a line. The image may be a degenerate line, that is, a single point.

• Show that the image of any linear surface is a linear surface. This generalizes the result that under a linear map the image of a subspace is a subspace.

• Linear maps preserve other linear ideas…

Show that any permutation, any reordering, $p$ of the numbers $1$, …, $n$, the map

can be done with a composition of maps, each of which only swaps a single pair of coordinates. Hint: you can use induction on $n$. (Remark: in the fourth chapter we will show this and we will also show that the parity of the number of swaps used is determined by $p$. That is, although a particular permutation could be expres…