# Proofs - Linear Algebra I MT22

> Source: https://ollybritton.com/notes/uni/prelims/mt22/linear-algebra/proofs/ · Updated: 2023-01-03 · Tags: proofs, linear-algebra, mt22

### Steinitz exchange lemma
Prove the Steinitz exchange lemma:

> Let $V$ be a vector space and let $X = \\{v_1, v_2, \ldots, v_n\\} \subseteq V$. Suppose that $u \in \langle X \rangle$ but that $u \notin \langle X \backslash \\{v_i\\} \rangle$ for some $i$. Let $Y = (X \backslash \\{v_i\\}) \cup \\{u\\}$. Then $\langle Y \rangle = \langle X \rangle$.

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Prove, using the Steinitz exchange lemma, that linearly independent sets are always smaller than or equal in size to spanning sets:

> Let $V$ be a vector space and let $S, T$ be finite subsets of $V$. If $S$ is linearly independent and $T$ spans $V$, then $|S| \le |T|$.

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Prove, by quoting a theorem:

> Let $V$ be a finite-dimensional vector space. All bases of $V$ are finite and of the same size.

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### The dimension formula
Prove the dimension formula:

> Let $U, W$ be subspaces of a finite dimensional vector space $V$. Then $\dim(U+W) + \dim(U \cap W) = \dim(U) + \dim(W)$.

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### The rank-nullity theorem
Prove the rank-nullity theorem:

> Let $V, W$ be finite dimensional vector spaces and $T: V \to W$ a linear transformation. Then $\text{rank}(T) + \text{nullity}(T) = \dim V$ .

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### Change of basis theorem
Prove an important precursor to the change of basis theorem,

> Let $U, V, W$ be finite-dimensional vector spaces with dimensions $m, n, p$ and ordered bases $\mathcal{U}, \mathcal{V}, \mathcal{W}$. Let $S : U \to V$ and $T : V \to W$ be linear.
> Let $A = {}_\mathcal{V}S_\mathcal{U}$ and $B = {}_\mathcal{W}T_\mathcal{V}$. Then $BA = {}_\mathcal{W}TS_\mathcal{U}$.

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Prove the change of basis theorem:

> Let $V$ be a finite-dimensional vector space with ordered bases $\mathcal{V}, \mathcal{V}'$.
> Let $W$ be a finite-dimensional vector space with ordered bases $\mathcal{W}, \mathcal{W}'$.
> Let $T : V \to W$ be a linear map. Then ${}_\mathcal{W'}T_\mathcal{V'} = ({}_\mathcal{W'}I_\mathcal{W})({}_\mathcal{W}T_\mathcal{V})({}_\mathcal{V}I_\mathcal{V'})$

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### Row rank and column rank
Prove, by appealing to the RREF, that the column rank and the row rank are equal.
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### Cauchy-Schwarz Inequality
Prove the Cauchy-Schwarz inequality:

> For $v, w$ in an inner product space $V$, then $|\langle v, w\rangle| \le ||v|| \text{ } ||w||$.

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### Triangle Inequality
Prove the triangle inequality in an inner product space:

> For $v, w$ in an inner product space $V$, then $||v+w|| \le ||v|| + ||w||$.

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