Question

12. Given that 2.V 3 which of the following choices will make S linearly independent? b) v= 1-15 24 0 d) v

Please show me the detail using matrix form.

Answer 1

if the matrix formed by the vectors have determinant as 0, then vectors are linear dependent.

$\\(b)\ A = \begin{bmatrix} 2 &1 &6 \\ -5 & -2 &-15 \\ 8& 3& 24 \end{bmatrix},\ det(A) =0 \\ \\so\ vectors\ are\ linearly\ dependent\\$

-5-2-151-2x-2-15 8 3 24 15 +5/ 24 =2x13 24 2 -15

$\\(c)\ A = \begin{bmatrix} 2 &1 &2 \\ -5 & -2 &-1 \\ 8& 3& 5 \end{bmatrix},\ det(A) =5\\ \\so\ vectors\ are\ linearly\ independent\\$

1 2 +8x 2 -1

$\\(d)\ A = \begin{bmatrix} 2 &1 &0 \\ -5 & -2 &0 \\ 8& 3& 0 \end{bmatrix},\ det(A) =0\\ \\as\ one\ column\ is\ completely\ 0\\ \\so\ vectors\ are\ linearly\ dependent\\$

so answer is option (c)

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