Determine if the columns of the matrix form a linearly independent set. Justify your answer. Choose the correct answer below. A. The columns of the matrix do form a linearly independent set because the set contains more vectors than there are entries in each vector. B. The columns of the matrix do form a linearly independent set because there are more entries in each vector than there are vectors in the set. C. The columns of the matrix do not form a linearly independent set because the set contains more vectors than there are entries in each vector. D. The columns of the matrix do not form a linearly independent set because there are more entries in each vector than there are vectors in the set.

The column of the matrix are [tex]\begin{bmatrix}1\\ -4\end{bmatrix}[/tex] , [tex]\begin{bmatrix}-4\\ 16\end{bmatrix}[/tex], [tex]\begin{bmatrix}4\\ 4\end{bmatrix}[/tex]

Now each of them are vectors in [tex]$IR^2$[/tex]. But [tex]$IR^2$[/tex] has dimensions of 2. But there are 3 column vectors, hence they are linearly dependent.

Therefore, the column of the given matrix does not form the [tex]\text{linearly independent set}[/tex] as the set contains [tex]\text{more vectors}[/tex] than there are entries in each vector.

Therefore, option (D) is correct.