202210110910

Type :Note Tags : Calc

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Alt

The $\text{Alt}$ function is of the type ${\mathcal{T}}^k(V)\to {\Lambda }^k(V)$ where ${\mathcal{T}}^k(V)$ is the set of all multilinear maps from $V^k\to \mathbb{R}$ ${\Lambda }^k(V)$ is the set of all skew-symmetric multilinear maps from $V^k\to \mathbb{R}$

$Al{t}^2:{\mathcal{T}}^2(V)\to {\Lambda }^2(V)$

let $B\in {\mathcal{T}}^2(V)$

$${\text{Alt}}^2(B)(v_1,v_2)=\frac{1}{2}(B(v_1,v_2)-B(v_2,v_1))$$

If $B\in {\Lambda }^2(V)$, Then ${\text{Alt}}^2(B)=B$

Similarly we can define:

${\text{Alt}}^k:{\mathcal{T}}^k\to {\Lambda }^k$

let $B\in {\mathcal{T}}^k(V)$

$${\text{Alt}}^k(B)(v_1,v_2\dots v_n)=\frac{1}{k!}\sum \text{sgn}(\sigma )B(v_{\sigma (1)},v_{\sigma (2)}\dots v_{\sigma (n)})$$

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Related Problems

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References

Exterior Algebra