5 votes

Intuition behind thinking of the transpose of matrices?

The transpose of a matrix is the matrix associated to the dual linear map. That is the main reason for the importance of the transpose in abstract linear algebra. Let $f : V \to W$ be a linear map ...
  • 33k
1 vote
Accepted

prove that $\det(I + tA^2)\ge 0$ for all real $t$

(Too long for a comment.) It doesn't seem clear why $\det(A^{-1} + tA)$ would be nonzero. It can be zero! Consider $A=\pmatrix{0&-1\\ 1&0}$ and $t=1$ for instance. A useful idea seems to be ...
1 vote
Accepted

How do I prove that $\dim F^\perp = n - \dim F + \dim F \cap \ker \phi$

In addition to what you have, we'll also apply the rank nullity theorem to $\Psi^*:F\to E^*$ and we'll use the fact that $\dim \text{im}(\Psi)=\dim \text{im}(\Psi^*)$. Note that, by symmetry, $\Psi^*:...
1 vote

Prove that for any nonsingular complex matrix $A$ and for any positive integer $k$, the equation $X^k = A$ has a solution

If $A$ is normal, then $A$ is unitarily similar to a diagonal matrix, i.e. a unitary matrix $P$ exists s.t. $PAP^{-1}=D$ with $D$ diagonal. If so, then $$ PX^{k}P^{-1} = (PXP^{-1})^{k} = PAP^{-1} = D =...
  • 9,782
1 vote

Which of the following quantities and operations constitute vector spaces?

Any vector in $X$ is of the form $(x,0)$ or $(0,y)$ for real numbers $x$ and $y$. Now take $x=y=1$ and add the vectors $(1,0)+(0,1)$ to get $(1,1)$. As you can see that this new vector is not in $X$ ...
  • 2,562
1 vote

How to show the number of distinct generator matrices of a linear code

A generator matrix $ G $ of an $ [n, k, d]$-code $ \mathcal C \subseteq \mathbb F_q^n $ is a matrix whose rows form a ($ \mathbb F_q $-linear) basis $ B = (b_1, \dotsc, b_k) $ of $ \mathcal C $. So ...
1 vote

Does the minimum singular value of a matrix smaller than that of its restricted one?

This is true if $A$ is a square or “fat” matrix, and false otherwise. A counterexample first: let $A=\pmatrix{0\\ 1}$ and $A_r$ be the first row of $A$. Then $\sigma_\min(A_r)=0<1=\sigma_\min(A)$. ...
  • 127k
1 vote
Accepted

Let $V$ be a vector space; prove the following statements:

You are correct about part $a$. For the first direction, if $M\cup \{v\}$ is linearly independent, then $v\notin \text{span(M)}$(it is more common to write span$(A)$ to denote the set of all linear ...
  • 2,562

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