Why does $\operatorname{rank}(\mathbf{X})$ equal $\operatorname{rank}(\mathbf{X^TX})$? What is $\dim(\mathbf{X^TX})$?
28565
Prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$
146541
Why $\dim U+\dim U^\perp=\dim V$?
1162
Why $\{\mathbf{0}\}$ has dimension zero?
100161
Is it true that $\dim(X) \leq \dim(X^{\ast})$ for every infinite dimentional Banach space $X$?
24072
Intuitive explanation of why $\dim\operatorname{Im} T + \dim\operatorname{Ker} T = \dim V$
53633
Given two subspaces $U,W$ of vector space $V$, how to show that $\dim(U)+\dim(W)=\dim(U+W)+\dim(U\cap W)$
87387
Linear equations given by $Ax = b$ have a solution if and only if $\operatorname{rank}(A|b) = \operatorname{rank}(A)$
152916
What is the gradient with respect to a vector $\mathbf x$?
129732
What is $\operatorname{Aut}(\mathbb{Z}_2\oplus \mathbb{Z}_2)$?
27821
A problem on rank of $A$ and $\operatorname{adj}(A)$
89595
What is the difference of \mathop, \operatorname and \DeclareMathOperator?
89177
$\operatorname{adj}(AB) = \operatorname{adj} B \operatorname{adj} A$
136909
Functional Analysis: The rank of an operator detemines if it's a compact operator.
11509
Relationship between $\operatorname{ord}(ab), \operatorname{ord}(a)$, and $\operatorname{ord}(b)$
147659
Counterexample to "if $f:A\to A$ induces the identity $\hat f:\operatorname{Spec}(A)\to\operatorname{Spec}(A)$, then $f=\operatorname{id}_A$"
52946
If $X$ is an orthogonal matrix, why does $X^TX = I$?
48660
Proof that determinant rank equals row/column rank