# Tagged Questions

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### How do you solve a linear transformation with no transformation matrix given?

I am stuck, I can't see how Tff was found with no transformation matrix. And now am being asked to find Tgg, help me http://oi60.tinypic.com/33yrplv.jpg
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### Number of Involutive Automorphisms on a Clifford Algebra

Let $V$ be a vector space with dimension $n$ and $q$ a quadratic form on $V$. How many involutive automorphisms are there in $\mathcal{Cl}(V,q)$?
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### Sum of the squares of the minors of a matrix with orthonormal column vectors = 1?

Let $A$ be an $m \times n$ ($n \leq m$) matrix with real entries and orthonormal column vectors. Claim: For $1 < k \leq n$, the sum of the squares of the $k\times k$ minors of $A$ is always $1$. ...
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### Trace of the $n$-th symmetric power of a linear map

Suppose $V$ is a vector space over $k$ and $\dim(V) = N$. Let $A \in\operatorname{End}(V)$. Let $\wedge^n A \in \operatorname{End}(\wedge^n V)$ where $\wedge^n$ is the $n$-th exterior power. I am ...
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### Bilinear Form - Proof

I have to prove that the mapping $f(x,y) = {\displaystyle \sum_{i=1} ^ {n} }{ \displaystyle \sum_{j=1}^{n} }x_iy_j{f}(e_i,e_j)$ is a bilinear form, that is, inter alia, the condition: ...
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### bilinear form - proof

I have to prove that the mapping $f(x,y)={\displaystyle \sum_{i=1}^{n}}{\displaystyle \sum_{j=1}^{n}}x_{i}y_{j}{f}(e_{i},e_{j})$ is a bilinear form, that is, inter alia, the condition: ...
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### Quick question: tensor product and dual of vector space

Recall that for a finite dimensional vector space $V$ we have the natural isomorphism $\phi :V^{*} \otimes V \rightarrow Hom(V,V)$ given by $\alpha \otimes v \mapsto (x \mapsto \alpha (x)v)$. Is ...
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### Computing distances between hyperspheres and sides of a hypercube?

Suppose you are given the $n$ dimensional hypersphere: $$\left(x_1 - \frac{1}{2}\right)^2 + \left(x_2 - \frac{1}{2}\right)^2 +\ldots+ \left(x_n - \frac{1}{2}\right)^2 = \frac{n}{4}$$ And the ...
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### Prove that $\phi_1 \wedge \cdots \wedge \phi_k (v_1, \cdots, v_k) = \frac{1}{k!}\det[\phi_i(v_j)].$

I have proved these two exercises: (1) Suppose that $T \in \Lambda^p(V^*)$ and $v_1, \ldots, v_p \in V$ are linearly dependent. Prove that $T(v_1, \ldots, v_p) = 0$ for all $T \in \Lambda^p(V^*)$. ...
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### Why the words “inner” and “outer” to designate products?

Does anyone know what's the rationale for using the adjectives inner and outer for certain algebraic products? Also, I've seen the term exterior algebra. Does the exterior here have anything to do ...
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### basis-independent isomorphism

Could some one give me some help with this proof? Given the hint, I still don't have a clue about how to proceed. Thanks.
### Representing a nonzero bilinear alternating form on a two-dimensional space by $\bigl(\begin{smallmatrix} 0&1\cr-1&0\end{smallmatrix}\bigr)$.
So I am having a little bit struggle with some question. I have a bilenear form $B:V\times V\to \Bbb F$. $B$ is not the zero form. $B$ is alternating meaning it is also skew-symmetric. I know that ...