##### Theorem5.1

(The Invertible Matrix Theorem)Let \(A\) be an \(n\times n\) matrix. Then the following conditions are equivalent:

- The columns of \(A\) are linearly independent.
- The columns of \(A\) are a spanning set for \(\mathbb{R}^n\).
- The colums of \(A\) are a basis for \(\mathbb{R}^n\).
- The rows of \(A\) are linearly independent.
- The rows of \(A\) are a spanning set for \(\mathbb{R}^n\).
- The rows of \(A\) are a basis for \(\mathbb{R}^n\).
- For any choice of vector \(b \in \mathbb{R}^n\), the system of linear equations \(Ax = b\) has a unique solution.
- \(A\) is invertible.
- The transpose \(A^T\) is invertible.
- \(\det(A) \neq 0\).
- \(\det(A^T) \neq 0\).