---
title: Quiz 3
tags: Linear algebra
GA: G-77TT93X4N1
---

# Quiz 3

* p.124: 1.1, 1.2, 1.3, 1.4, 1.7
* Suppose $V$ is an inner product space, $W$ is a vector space and $T:V\to W$ is an isomorphism. For $w_1, w_2\in W$, define
$$
\langle w_1, w_2\rangle:=\langle T^{-1}w_1, T^{-1}w_2\rangle,
$$
where the right-hand side involves the given inner product on $V$. Prove that this defines an inner product on $W$.
* Show that, for real ${\bf u}$ and ${\bf v}$,
$$
\langle{\bf u}+{\bf v}, {\bf u}-{\bf v}\rangle = \|{\bf u}\|^2 - \|{\bf v}\|^2.
$$
* Suppose $V$ is an inner product space, $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$ are both inner products defined on $V$. Show that
$$
\langle{\bf u}, {\bf v}\rangle = \langle{\bf u}, {\bf v}\rangle_1 + \langle{\bf u}, {\bf v}\rangle_2
$$
defines an inner product on $V$.

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# Quiz 04/10

* [quiz0410](https://drive.google.com/file/d/1lrzcnBCX7w7MA9a7A_Bc2u3N69e920Ai/view?usp=sharing)