Notes - Linear Algebra I MT22, Chapter 8

[[Course - Linear Algebra I MT22]]^U
Content covered:
- Prove that if $A$ is the Gram matrix representing a bilinear form $B$ then $B(v, w) = xAy^\intercal$ (Prop. 197).
- Prove the triangle inequality for norms. (Prop. 205)
- Prove the Cauchy-Schwarz inequality. (Prop. 206)
- Prove that an orthogonal map is an isometry of an inner product space. (Prop. 212)

What does it mean for a linear transformation $T$ to be orthogonal?

\[\langle T(v), T(w)\rangle = \langle v, w\rangle\]

for all $v$, $w$.

What does it mean for a set $\{v _ 1, v _ 2, \ldots, v _ k\}$ to be an orthonormal set, in notation?

\[\langle v_i, v_j\rangle = \begin{cases} 1, & \text{ if } i =j \\\\ 0, & \text{ otherwise} \end{cases}\]

A set $\{v _ 1, v _ 2, \ldots, v _ k\}$ is orthonormal if

\[\langle v_i, v_j\rangle = \begin{cases} 1, & \text{ if } i =j \\\\ 0, & \text{ otherwise} \end{cases}\]

for all $i$ and $j$. What does this mean in English?:: All vectors are of unit length and mutually perpindicular.

In terms of a metric $d$ given by an inner product space, what does it mean for a linear transformation $T$ to be an isometry?

\[d(v, w) = d(T(v), T(w))\]

What is true about any orthogonal map in an inner product space?

It is an isometry.