MATH 3231 Class Notes 02-21-2025

notering-theoryvector-spaces

Vector Spaces

Defined Support

Defined Linear Map

Defined Linear Isomorphism

Theorem - Every Vector Space has a Basis

Theorem

If there exists a vector space $V$ over $k$, with bases $\mathcal{B}=(v_i{)}_{i\in I}$ and $\mathcal{C}=(w_j{)}_{j\in J}$, then there exists a bijection $f:I\to J$.

Corollary

${\dim }_{k}V\stackrel{def}{=}$ the cardinality of any basis of $V$.

Definition

For a vector space $V$ with basis $\mathcal{B}=(v_i{)}_{i\in I}$, we get $T:k^{\oplus I}\to V$ induced isomorphism, and we define $[v{]}_{\mathcal{B}}\stackrel{def}{=}{T}^{-1}(v)$, the coordinates of $v$ with respect to $\mathcal{B}$.

Definition

If $T:V\to V$ a linear operator, and $\mathcal{B}=(v_i{)}_{i\in I}$ is a basis, where $\#I=n$, then there exists a unique $n\times n$ matrix $[T{]}_{\mathcal{B}}=[T{]}_{\mathcal{B}}\cdot [v{]}_{\mathcal{B}}$, defined by,

$$[T{]}_{\mathcal{B}}=\left[\begin{array}{ccc}⋮& & ⋮\\ [T(v_1{)}_{\mathcal{B}}& \cdots & [T(v_n){]}_{\mathcal{B}}]\\ ⋮& & ⋮\end{array}\right].$$

Definition

$T:V\to V$ a linear operator. $P(x)\in k[x]$ with $P(x)=a_n{x}^n+\cdots +a_0$. We can define $P(T):V\to V$ a linear operator given by

$$P(T)=a_n{T}^n+\cdots +a_{1}T+a_j\cdot I_v$$

Where $I_v$ is the identity function such that $I_v(v)=v$ for all $v\in V$.

Defined Characteristic Polynomial

Remark

If $\dim V=n$ and $\mathcal{B}=v_1,\dots ,v_n$ and $\mathcal{C}=w_1,\dots ,w_n$ are bases of $V$, and $T$ is a linear operator on $V$, then $[T{]}_{\mathcal{B}},[T{]}_{\mathcal{C}}$ are “similar” matrices.

Remark

For a linear operator $T$ on $V$, where $\dim V=n$,

$$k[x]\stackrel{\varphi }{\to }{\mathrm{end}}_k(V)$$

A morphism of rings, also a linear transformation with

$$P(x)\mapsto P(T)\text{ as before}$$