Prove that a linear operator $T$ on a finite-dimensional vector space is invertible if and only if zero is not an eigenvalue of $T$.
52208
Show that a matrix $A$ is singular if and only if $0$ is an eigenvalue.
124942
Prove that an upper triangular matrix is invertible if and only if every diagonal entry is non-zero.
62303
For $V$ vector space of dimension $n$ under $\mathbb C$ and $T: V \to V $ linear transformation , show $V= \ker T^n \oplus $ Im $T^n$
48906
Prove that T is diagonalizable if and only if the minimal polynomial of T has no repeated roots.
11715
Prove that if $\langle Tx,x\rangle =0$ for all $x \in X$, then $T = 0$
78307
Under what conditions is $AA^T$ invertible?
93121
How to prove there exists a linear transformation?
10001
Prove that the only eigenvalue of a nilpotent operator is 0?
2439
If a linear operator has an adjoint operator, it is bounded
137920
Proving an equation with invertible matrices
103839
Proving that $T^{2} = T$ for a linear operator on $W$ implies that $V = \operatorname{Null} T \oplus \operatorname{Range} T$
35345
How do i prove invertibility of this linear transformation
72240
Prove that if a linear operator is continuous, then it is bounded.
144019
A strictly positive operator is invertible
109217
Invertibility of the Product of Two Matrices
157269
If $\alpha_{1},\ldots,\alpha_{m}$ are vectors different from zero vector, then there is a linear functional $f$ such that $f(\alpha_{i})\neq 0$
91022
How do I prove that a subspace of a vector space $X$ is the null space of some linear functional on $X$?