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Bases and dimensions of exterior powers of $R^{2}$ , $R^{3}$ , and $R^{4}$

Example

In the standard bases, the exterior powers have the following bases and dimensions. For $R^{2}$ : $Λ^{0} = R$ (basis $1$ ), $Λ^{1}$ has basis $e_{1}, e_{2}$ , $Λ^{2}$ has basis $e_{1} \land e_{2}$ , and $Λ^{k} = 0$ for $k > 2$ . For $R^{3}$ : $Λ^{0} = R$ ; $Λ^{1}$ has basis $e_{1}, e_{2}, e_{3}$ ; $Λ^{2}$ has basis $e_{1} \land e_{2}, e_{1} \land e_{3}, e_{2} \land e_{3}$ ; $Λ^{3}$ has basis $e_{1} \land e_{2} \land e_{3}$ ; and $Λ^{k} = 0$ for $k > 3$ . For $R^{4}$ : $Λ^{0} = R$ ; $Λ^{1}$ has basis $e_{1}, e_{2}, e_{3}, e_{4}$ ; $Λ^{2}$ has basis $e_{1} \land e_{2}, e_{1} \land e_{3}, e_{1} \land e_{4}, e_{2} \land e_{3}, e_{2} \land e_{4}, e_{3} \land e_{4}$ ; $Λ^{3}$ has basis $e_{1} \land e_{2} \land e_{3}, e_{1} \land e_{2} \land e_{4}, e_{1} \land e_{3} \land e_{4}, e_{2} \land e_{3} \land e_{4}$ ; $Λ^{4}$ has basis $e_{1} \land e_{2} \land e_{3} \land e_{4}$ ; and $Λ^{k} = 0$ for $k > 4$ .

Facts & Assumptions

Given: The standard bases of $R^{2}$ , $R^{3}$ , and $R^{4}$ .

[L1]

The increasing-index wedges of an ordered basis form a basis of $Λ^{k} V$ (Increasing-index wedges of a basis form a basis of $Λ^{k} V$ ).

[L2]

$\dim Λ^{k} V = (\binom{n}{k})$ (If $\dim V = n$ , then $\dim Λ^{k} V = (\binom{n}{k})$ ).

[L3]

$Λ^{k} V = 0$ for $k > \dim V$ (If $k > \dim V$ , then $Λ^{k} V = 0$ ).

Verification

technique · direct

1.1

By [L1], the listed families are exactly the increasing-index wedges of the standard bases, so they are bases of the corresponding exterior powers.

L1given

1.2

By [L2], the dimensions are the binomial counts: $1, 2, 1$ for $R^{2}$ ; $1, 3, 3, 1$ for $R^{3}$ ; and $1, 4, 6, 4, 1$ for $R^{4}$ , matching the sizes of the listed families.

1.3

By [L3], every degree above the top vanishes, so no further nonzero pieces appear.

2.1

Steps 1.1, 1.2 and 1.3 verify all displayed bases, dimensions, and vanishing statements.

step 1.1step 1.2step 1.3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Keith Conrad, Exterior Powers (standard reference, not scraped)

Bases and dimensions of exterior powers of R2R2, R3R3, and R4R4