Are the vectors [−24],[7−2][−24],[7−2] and [3−6][3−6] linearly independent? Yes No

To determine whether the vectors $\begin{bmatrix} -24 \\ 7 \\ -2 \end{bmatrix}$, $\begin{bmatrix} 3 \\ -6 \end{bmatrix}$, and the vector $\begin{bmatrix} 3 \\ -6 \end{bmatrix}$ are linearly independent, we can use the concept of linear independence in vector spaces.

Step 1: Break Down the Problem

We have two vectors and we want to check if they are linearly independent. To check for linear independence, we verify if the only solution to the equation: $c_1 \begin{bmatrix} -24 \\ 7 \\ -2 \end{bmatrix} + c_2 \begin{bmatrix} 3 \\ -6 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is $c_1 = 0$ and $c_2 = 0$.

Step 2: Relevant Concepts

For the vectors to be linearly independent, the determinant of the matrix formed by the vectors should not be zero. If there are more vectors than dimensions represented, they can’t be linearly independent.

Step 3: Analysis and Detail

Let's set the matrix:

$A = \begin{bmatrix} -24 & 3 \\ 7 & -6 \\ -2 & 0 \end{bmatrix}$

We can find out if these vectors are linearly independent by reducing the matrix to row-echelon form.

1. Form the augmented matrix: $\begin{bmatrix} -24 & 3 & 0 \\ 7 & -6 & 0 \\ -2 & 0 & 0 \end{bmatrix}$
2. Perform row operations to simplify the matrix.
   • Divide the first row by -24 (to make a leading coefficient 1).
   • Eliminate the first variable from other rows.

After performing the necessary row operations, if we can find a row of the form [0 0 | d] (where d is not zero), it indicates linear dependence.

Step 4: Verify and Summarize

After the row-reduction, if the number of pivot columns equals the number of the original vectors, they are independent. If less, they are dependent.

Analysis results: After computation, if you find that one of the vectors can be expressed as a linear combination of the others, they are linearly dependent, otherwise independent.

Final Answer

Since the provided vectors span a space that exceeds their count, they are linearly dependent. Thus, the answer is: No, they are not linearly independent.