Invertible functions and their properties

Question

If an n × n matrix A is singular, then the columns of A must be linearly independent. Is this true?

Invertible functions must be bijective

Invertible functions must have square matrices

Invertible functions must span R^n

Also, am I missing some other must conditions of invertible or singular functions?

What's special about invertible functions anyway?

Answer 1

If an $n \times n$ matrix $A$ is singular, then the columns of $A$ must be linearly independent. Is this true?

No, it's exactly the opposite.

Invertible functions must be bijective

Yes.

Invertible functions must have square matrices

Invertible linear functions can be represented by square (nonsingular) matrices.

Invertible functions must span $\mathbb{R}^n$.

Assuming that they are linear, and their codomain is $\mathbb{R}^n$, then yes.

Also, am I missing some other must conditions of invertible or singular functions?

Determinant, eigenvalues,...

What's special about invertible functions anyway?

They have inverse.