Proofs - Linear Algebra I MT22

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 $\langle Y \rangle = \langle X \rangle$.

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 $ \vert S \vert \le \vert T \vert $.

Prove, by quoting a theorem:

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

Prove the dimension formula:

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

Prove the rank-nullity theorem:

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

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 $BA = {} _ \mathcal{W}TS _ \mathcal{U}$.

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 ${} _ \mathcal{W’}T _ \mathcal{V’} = ({} _ \mathcal{W’}I _ \mathcal{W})({} _ \mathcal{W}T _ \mathcal{V})({} _ \mathcal{V}I _ \mathcal{V’})$

Prove, by appealing to the RREF, that the column rank and the row rank are equal.

Prove the Cauchy-Schwarz inequality:

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

Prove the triangle inequality in an inner product space:

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