Lecture 39

Chapter IX Multilinear Algebra and Determinants

Exterior Powers ?A

Definitions ?.1

Let $V$ be a vector space, the n-th exterior power of $V$ denoted $\wedge^m V$ is a vector space formed by finite linear combination of expression of the form $v_1\wedge v_2\wedge\dots \wedge v_m$. subject to relations:

1. $c(v_1\wedge v_2\wedge\dots \wedge v_m)=(cv_1)\wedge v_2\wedge\dots \wedge v_m$
2. $(v_1+w_1)\wedge v_2\wedge\dots \wedge v_m=(v_1\wedge v_2\wedge\dots \wedge v_m)+(w_1\wedge v_2\wedge\dots \wedge v_m)$
3. Swapping two entires in ($v_1\wedge v_2\wedge\dots \wedge v_m$) gives a negative sign.

Example:

$\wedge^2\mathbb{R}^3$

Theorem ?.2

$0\wedge v_1\wedge\dots\wedge v_m=0$

Proof:

Theorem ?.3

$v_1\wedge v_1\wedge\dots\wedge v_m=0$

Proof:

swap $v_1$ and $v_1$.

Theorem ?.4

$v_1\wedge v_2\wedge\dots\wedge v_m\neq 0$ if and only if $v_1,\dots ,v_m$ are linearly independent.

Proof:

We first prove forward direction,

Suppose $v_1,\dots, v_m$ are linearly dependent then let $a_1v_1+\dots +a_nv_m=0$ be a linear dependence. Without loss of generality. $a\neq 0$ then consider

reverse is the similar.

Theorem ?.5

If $v_1,\dots v_n$ forms a basis for $V$, then expressions of the form $v_{i_1}\wedge\dots \wedge v_{i_m}$ for $1\leq i_1\leq i_m\leq n$ forms a basis of $\wedge^m V$

Proof:

Spanning: Let $u_1\wedge\dots \wedge u_m\in \wedge^m V$ where $u_1=a_{1,1}v_1+\dots+a_{1,n}v_n,u_m=a_{m,1}v_1+\dots+a_{m,n}v_n$

Expand: then we set expressions of the form $\plusmn c(v_{i_1}\wedge \dots \wedge v_{i_m})$. Let $A=(a_{i,j})$ , $c$ is the $m\times m$ minor for the columns $i_1,..,i_m$.

Corollary ?.6

Let $n=dim\ V$ then $dim\ \wedge^n v=1$

Note $dim\ \wedge^m V=\begin{pmatrix} n\\m \end{pmatrix}$

Proof: Chose a basis $v_1,...,v_n$ of $V$ then $v_1\wedge \dots \wedge v_n$ generates $\wedge^n v$.

Definition ?.7

Let $T\in\mathscr{L}(V)$, $n=dim\ V$ define $det\ T$ to be the unique number such that for $v_1\wedge\dots\wedge v_n\in \wedge^n V$. $(Tv_1\wedge\dots\wedge Tv_n)=(det\ T)(v_1\wedge \dots \wedge v_n)$

Theorem ?.8

1. Swapping columns negates the determinants
2. $T$ is invertible if and only if $det\ T\neq 0$
3. $det(ST)=det(S)det(T)$
4. $det(cT)=c^n det(T)$