If the column of matrix a are linearly independent, then

If A's columns are linearly independent, then it is invertible.

1. What does it mean if the columns of a matrix are linearly independent?
2. What happens if a columns of a matrix linearly dependent?
3. Why the columns of an nxn matrix A are linearly independent when A is invertible?
4. When matrix is linearly independent?

What does it mean if the columns of a matrix are linearly independent?

The columns of matrix A are linearly independent if and only if the equation Ax = 0 has only the trivial solution. Fact. A set containing only one vector, say v, is linearly independent if and only if v = 0. This is because the vector equation x1v = 0 has only the trivial solution when v = 0.

What happens if a columns of a matrix linearly dependent?

The columns of A are linearly dependent if and only if Ax = 0 has a non-zero solution. The columns of A are linearly dependent if and only if A has a non-pivot column. The columns of A are linearly independent if and only if Ax = 0 only for x = 0.

Why the columns of an nxn matrix A are linearly independent when A is invertible?

Explain why the columns of an n by n matrix are linearly independent when A is invertible. If A is invertible, then the equation Ax=0 has a unique solution, the trivial solution, so the columns of A must be linearly independent.

When matrix is linearly independent?

If the determinant is not equal to zero, it's linearly independent. Otherwise it's linearly dependent. Since the determinant is zero, the matrix is linearly dependent.