Question

Question: Are the columns of an invertible matrix linearly independent?

Short answer

Yes, the columns of an invertible matrix are linearly independent.

Step-by-step solution

Definition of Linearly independent

Linearly independent is defined as the property of a set having no linear combination of its elements equal to zero when the coefficients are taken from a given set unless the coefficient of each element is zero.

Kernel of invertible

For vectors $\overrightarrow{v_1},\overrightarrow{v_1},...,\overrightarrow{v_m},$, we know that the following statements are equivalent $-\backslash par1$. The vectors $\overrightarrow{v_1},\overrightarrow{v_1},...,\overrightarrow{v_m},$are linearly independent $-/par2$$.\ker \left(A\right)=\left\{0\right\}$, where is the given vectors as their columns.

Also, the kernel of an invertible matrix is zero.

The final answer

A is invertible $\backslash par\iff \ker \left(\mathrm{A}\right)=\left\{0\right\}\backslash \mathrm{par}\iff$Column vectors of are linearly independent.

Therefore, the columns of an invertible matrix are linearly independent.