32 - Properties and Types of a Linear Map Matrix

Zero Map Matrix

Let $V$ be a $n$-dimensional vector space over $\mathbf{F}$ with ordered basis $\mathcal{V}$
Let $W$ be a m-dimensional vector space over $\mathbf{F}$ with ordered basis $\mathcal{W}$

Let ${}_{\mathcal{W}}{0}_{\mathcal{V}}$ be the matrix represented by the zero map
Then

$${}_{\mathcal{W}}{0}_{\mathcal{V}}=0_{m\times n}\text{ for any choice of }\mathcal{V}\text{ and }\mathcal{W}$$

Identity Map Matrix

Let $V$ be a $n$-dimensional vector space over $\mathbf{F}$ with ordered basis $\mathcal{V}$

Let ${}_{\mathcal{V}}{I}_{\mathcal{V}}$ be the matrix represented by the identity map
Then

$${}_{\mathcal{V}}{\text{id}}_{\mathcal{V}}=I_n\text{ for any choice of }\mathcal{V}$$

Linear Property of a Linear Map Matrix

Let $V$ be a $n$-dimensional vector space over $\mathbf{F}$ with ordered basis $\mathcal{V}$
Let $W$ be a m-dimensional vector space over $\mathbf{F}$ with ordered basis $\mathcal{W}$

If $S:V\to W,T:V\to W$ are linear and for $\alpha ,\beta \in \mathbf{F}$ then

$${}_{\mathcal{W}}(\alpha S+\beta T{)}_{\mathcal{V}}=\alpha {(}_{\mathcal{W}}{S}_{\mathcal{V}})+\beta {(}_{\mathcal{W}}{T}_{\mathcal{V}})$$

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Properties of Linear Maps

Let $A={}_{\mathcal{V}}{T}_{\mathcal{V}}$ and $B={}_{\mathcal{W}}{T}_{\mathcal{W}}$ be matrices represented $T$ with respect to two bases so that

$$A=P^{-1}BP\text{ for some invertible }P$$

1. $A$ is invertible if and only if $B$ invertible
2. Trace of $A$ equals the trace of $B$ (follows from identity $\mathrm{trace}(MN)=\mathrm{trace}(NM)$)
3. A functional identity satisfied by $A$, such as $A^2=A$, is also satisfied by $B$
4. Determinant of $A$ equals the determinant of $B$
5. Eigenvalues of $A$ equals the eigenvalues of $B$

Note that $(4)$ and $(5)$ will be formally defined in Linear Algebra II