4. (18 pts) Suppose that ϕ is an n-permutation, and that Pϕ is its corresponding permutation mat

So the question is P is equal to P1, P2, rest less as P K, where each P i is an elementary permu

For exercise 24, we have C -0, C -s -1, C -s -2, C -s -s -3, C -2, C -2, C -1, C -s -0, C -s -0,

So here since x is in v and y is in v. So according to the linear space, we know x minus 1 is al

Hi in the given problem we are given with the permutation matrix P so that is 0 001 in each row

we have sigma inverse of sigma that is equal to an identity permutation and the epsilon should b

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4. (18 pts) Suppose that $\phi$ is an $n$-permutation, and that $P_{\phi}$ is its corresponding permutation matrix.\ Let $e_n = (e_1, e_2, \dots, e_n)$ be the standard basis for $\mathbb{R}^n$. Show that $P_{\phi}e_{\phi(i)} = e_i$.\ Given a vector space $V$, we can define the $k^{th}$ exterior power of $V$, denoted $\bigwedge^k V$, as the vector space spanned by expressions of the form $\vec{v}_1 \wedge \vec{v}_2 \wedge \dots \wedge \vec{v}_k$ where $\vec{v}_i \in V$. Such expressions are sometimes called multivectors. This wedge product, "$\wedge$", satisfies the following axioms: $\bullet$ Associativity: $(\vec{v}_1 \wedge \vec{v}_2) \wedge \vec{v}_3 = \vec{v}_1 \wedge (\vec{v}_2 \wedge \vec{v}_3)$.\ $\bullet$ Distrbutivity: $\vec{v} \wedge (\vec{u}_1 + \vec{u}_2) = (\vec{v} \wedge \vec{u}_1) + (\vec{v} \wedge \vec{u}_2)$.\ $\bullet$ Anticommutivity: $\vec{v} \wedge \vec{u} = -\vec{u} \wedge \vec{v}$.\ $\bullet$ Compatibility with scalar product: $(k\vec{v}) \wedge \vec{u} = \vec{v} \wedge (k\vec{u})$ where $k \in \mathbb{R}$.\ Because of the third property, $\vec{v} \wedge \vec{v} = 0$ for any vector $\vec{v}$. Because of the fourth property, we can write both sides of the equation as $k(\vec{v} \wedge \vec{u})$.

          4. (18 pts) Suppose that $\phi$ is an $n$-permutation, and that $P_{\phi}$ is its corresponding permutation matrix.\
Let $e_n = (e_1, e_2, \dots, e_n)$ be the standard basis for $\mathbb{R}^n$. Show that $P_{\phi}e_{\phi(i)} = e_i$.\
Given a vector space $V$, we can define the $k^{th}$ exterior power of $V$, denoted $\bigwedge^k V$, as the vector space spanned by expressions of the form
$\vec{v}_1 \wedge \vec{v}_2 \wedge \dots \wedge \vec{v}_k$
where $\vec{v}_i \in V$. Such expressions are sometimes called multivectors. This wedge product, "$\wedge$", satisfies the following axioms:
$\bullet$ Associativity: $(\vec{v}_1 \wedge \vec{v}_2) \wedge \vec{v}_3 = \vec{v}_1 \wedge (\vec{v}_2 \wedge \vec{v}_3)$.\
$\bullet$ Distrbutivity: $\vec{v} \wedge (\vec{u}_1 + \vec{u}_2) = (\vec{v} \wedge \vec{u}_1) + (\vec{v} \wedge \vec{u}_2)$.\
$\bullet$ Anticommutivity: $\vec{v} \wedge \vec{u} = -\vec{u} \wedge \vec{v}$.\
$\bullet$ Compatibility with scalar product: $(k\vec{v}) \wedge \vec{u} = \vec{v} \wedge (k\vec{u})$ where $k \in \mathbb{R}$.\
Because of the third property, $\vec{v} \wedge \vec{v} = 0$ for any vector $\vec{v}$. Because of the fourth property, we can write both sides of the equation as $k(\vec{v} \wedge \vec{u})$.

4. (18 pts) Suppose that ϕ is an n-permutation, and that Pϕ is its corresponding permutation matrix.Let en = (e1, e2, …, en) be the standard basis for ℝ^n. Show that Pϕeϕ(i) = ei.Given a vector space V, we can define the k^th exterior power of V, denoted ⋀^k V, as the vector space spanned by expressions of the form
v⃗1 ∧v⃗2 ∧…∧v⃗k
where v⃗i ∈ V. Such expressions are sometimes called multivectors. This wedge product, "∧", satisfies the following axioms:
∙ Associativity: (v⃗1 ∧v⃗2) ∧v⃗3 = v⃗1 ∧ (v⃗2 ∧v⃗3).∙ Distrbutivity: v⃗∧ (u⃗1 + u⃗2) = (v⃗∧u⃗1) + (v⃗∧u⃗2).∙ Anticommutivity: v⃗∧u⃗ = -u⃗∧v⃗.∙ Compatibility with scalar product: (kv⃗) ∧u⃗ = v⃗∧ (ku⃗) where k ∈ℝ.Because of the third property, v⃗∧v⃗ = 0 for any vector v⃗. Because of the fourth property, we can write both sides of the equation as k(v⃗∧u⃗).

Added by Traci J.

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