Linear Algebra I MT22, Chapter 8
- Course - Linear Algebra MT22U
- 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)
Flashcards
@Define what it means for a linear transformation $T$ to be orthogonal.
\[\langle T(v), T(w)\rangle = \langle v, w\rangle\]
for all $v$, $w$.
@Define what it means 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 perpendicular.
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.