Question

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.

Answer 1

Th matrix is missing. The matrix is :

$\begin{bmatrix}1 &-4 &4 \\ -4 &16 & 4 \end{bmatrix}$

Solution :

The column of the matrix are $\begin{bmatrix}1\\ -4\end{bmatrix}$ , $\begin{bmatrix}-4\\ 16\end{bmatrix}$, $\begin{bmatrix}4\\ 4\end{bmatrix}$

Now each of them are vectors in $IR^2$. But  $IR^2$ 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 $\text{linearly independent set}$ as the set contains $\text{more vectors}$ than there are entries in each vector.

Therefore, option (D) is correct.