Question

If the kernel of a matrix A consists of zero vector only, then the column vectors of A must be linearly independent.

Short answer

The above statement is true.

If the kernel of a matrix A consists of zero vector only, then the column vectors of A must be linearly independent.

Step-by-step solution

Definition of a kernel of a matrix

If A is a matrix of order m × n then the kernel of the matrix A, denoted by ker(A), is defined as -

$Ker\left(A\right)=\left\{x\in V:AX=0\right\}$

Mentioning the concept

Step 2: Mentioning the concept

For any matrix, A of order m × n, the equation AX = 0 has the trivial solution always.

Therefore, the kernel of matrix A always contains a zero vector.

Now, let us assume AX = 0 has only one unique solution ‘X’.

Let $C_1,C_2,...C_n$are the columns of matrix A.

Let the column vectors $C_1,C_2,...C_n$are linearly dependent. Then, by definition of linearly dependent vectors, there would exist scalars$a_1,a_2,...a_n$ , not all of them zero, such that-

$a_1{C}_1+a_2{C}_2+...+a_n{C}_n=0$

This says, AY = 0 for $Y={\left[a_1,a_2,...a_n\right]}^T$, which is a non-zero vector.

Thus, we have another solution y = X + Y:

A(X+Y) = A(X) + A(Y) = 0 + 0 = 0

which cannot be possible.

So, no such Y exists and the column vectors $C_1,C_2,...C_n$are linearly independent.

Final Answer

Since, for the given column vectors $C_1,C_2,...C_n$, we didn’t get any scalars $a_1,a_2,...a_n$, not all of them 0, such that-

$a_1{C}_1+a_2{C}_2+...+a_n{C}_n=0$

Thus, the column vectors $C_1,C_2,...C_n$are linearly independent.

Hence, if the kernel of a matrix A consists of zero vector only, then the column vectors of A must be linearly independent.