# Tag Info

### 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 ...
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1 vote
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### 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

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, $\Psi^*:... • 515 1 vote ### Prove that for any nonsingular complex matrix$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 ...
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