# Tag Info

### [Vector spaces ]Why does the matrix of a linear transformation from $V$ ( of dim $n$) to $W$ ( of dim $p$) have $p$ rows?

As others have already commented, it's because $W$ is $p$-dimensional, so each image vector $f(b_i)$ has $p$ coordinates with respect to the basis $c_1,\ldots,c_p$, and therefore $p$ rows in the ...
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### Construction of the matrix of a linear map between two vector spaces : difficulty in understanding how an expression represents a coordinate vector.

If you accept that the $[F(v_i)]_{B_W}$ are column vectors (one for every i from $1$ to $n$) then the sigma expression is just a sum of scaled column vectors, i.e. a column vector.
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### Why does the Gaussian-Jordan elimination works when finding the inverse matrix?

Let me make it concrete in the following example(@pauly-b has written a much more beautiful answer above). Suppose that 4 apples(any two are the same price) and 3 bananas(any two are the same price) ...
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### Determine axis of rotation from 4x4 transformation matrix

If you are using a 4x4 homogeneous transformation matrix, the point is either located in the last column or the bottom row, depending on conventions. Note that the element at 4,4 is typically 1 and ...
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Accepted

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### Show that $A \cdot \textbf{0}_{n \times p}=\textbf{0}_{m \times p}$

$A \cdot \textbf{0}_{n \times p}=\textbf{0}_{m \times p}$ For all $v\neq 0$ $A\cdot \textbf{0}_{n \times p} v =0$ and then use the uniqueness of the zero element.
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### What is the matrix corresponding to the quadratic form $x^2 + y^2 + z^2$?

When you want to find the matrix of a quadratic form, as with linear maps you merely need to evaluate it at basis vectors. But for quadratic forms you need to evaluate it at pairs of basis vectors. In ...
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### What is the matrix corresponding to the quadratic form $x^2 + y^2 + z^2$?

$$A = \begin{pmatrix}1 & 0 & 0 \\ 0& 1&0 \\ 0 & 0 & 1\end{pmatrix}$$
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1 vote

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Accepted

### Circulant-like determinant

Let $\zeta\in\mathbb{C}$ with $\zeta^{2n}=1=-\zeta^n$. I think, $v=(1,\zeta,\ldots,\zeta^{n-1})$ is an eigenvector with eigenvalue $\sum_{k=0}^{n-1}a_k\zeta^k$. Now let $\zeta:=e^{\pi i/n}$. This ...
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### Exercise 10, Section 3.4 of Hoffman’s Linear Algebra

If you're familiar with the idea of diagonalization, then since $S$ satisfies $S^2=S$ and is an operator on $\mathbb{R}^2$, the characteristic polynomial of $S$ is then $p(z)=z^2-z$. Since $S\neq I$ ...
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1 vote

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### Hodge star operator : inner product of $k$-forms independent of orthogonal frames

Follow Nicholas Todoroff's hint (generalized Cauchy-Binet Formula), we have \sum_{I}(\sum_{J}a_J Q(I,J))(\sum_{J }b_J Q(I,J)))=\sum_{I }\sum_{J,K}a_Jb_KQ(I,J)Q(I,K)=\sum_{J,K}\sum_{I }a_Jb_KQ(I,J)Q(...
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