Question

The column vectors of a 5×4 matrix must be linearly dependent.

Short answer

The above statement is false.

If A is a matrix of order$5\times 4$ , then the column vectors of A need not be linearly dependent.

Step-by-step solution

Concept of linearly dependence and linearly independence matrix

If A is a matrix of order $m\times n$, then

$Rank\left(A\right)⩽min\left\{m,n\right\}$

The numbers of unknown variables in an $m\times n$matrix is n.

If the rank of a matrix A is equal to the number of unknown variables of the given matrix A then the matrix A is linearly independent.

Otherwise, linearly dependent.

To find whether the given matrix is linearly dependent or linearly independent

Here, we have a matrix A of order$5\times 4$ .

Then,

$Rank\left(A\right)⩽min\left\{5,4\right\}$

$\Rightarrow Rank\left(A\right)⩽4$

The number of unknown variables in a$5\times 4$ matrix is 4.

$\Rightarrow Rank\left(A\right)=Thenumberofunknownvariables$

$\Rightarrow ThematrixAislinearlyindependent.$

Thus, the given matrix A is linearly independent.