Let $A$ be a subspace of $M_{n\times n}(F)$, prove that if $\dim(A)> n^2-n$ , $\exists$ invertible matrix $M \in A$.
2109
Max dimension of a subspace of singular $n\times n$ matrices
138069
Proof that $\bigcap_{a\in A} \bigg(\, \bigcup_{b\in B} F_{a,b} \, \bigg) = \bigcup_{f\in ^AB} \bigg(\, \bigcap_{a \in A} F_{a,f(a)}\,\bigg) $
150720
Given that $W$ is a subspace of $V$, prove: $\dim W=\dim V \implies W=V$
128760
If $A$ and $B$ two $n\times n$ matrix such that $AB=BA$, then $\dim(A\cap B)$?
95513
Proving $W$ is a subspace of $P_{2}$
55031
Show any matrix $A_{n\times n}$ can be written as sum of two nonsingular matrices
61103
Prove that $\mathbb{|Q| = |Q\times Q|}$
19396
For subspaces, if $N\subseteq M_1\cup\cdots\cup M_k$, then $N\subseteq M_i$ for some $i$?
1799
How many k-dimensional subspaces there are in n-dimensional vector space over $\mathbb F_p$?
57649
Prove that for every $f$, a bilinear form, there exists a basis ${v_1,...,v_n}$ so that $f(v_i,v_j) = -f(v_j,v_i)$
66729
For a square matrix $A$ presence of $n$ linearly independent eigenvectors implies full rank, thereby invertible?
75211
Theorem about positive matrices
25488
Let $A,B$ be $m\times m$ matrices such that $AB$ is invertible. Show $A,B$ invertible.
152817
Proving $\sup B=\inf A$
115582
Prove that if a product$ AB$ of $n\times n$ matrices is invertible, so are the factors $A$ and $B$.
80916
How to prove that if $A\subseteq A\times A$, then $A=\varnothing$?
127127
Prove condition number of a invertible matrix is atleast one