6.1

Question 1

Are these vectors linearly dependent ?
$\Bigg\{ \begin{pmatrix} -1 \\ 2 \\ 2 \\ \end{pmatrix} , \begin{pmatrix} -2 \\ -2 \\ -2 \\ \end{pmatrix} , \begin{pmatrix} 0 \\ 0 \\ 0 \\ \end{pmatrix} \Bigg\}$

yes

no

Question 2

What are $\lambda_i$’s value ?
$\begin{pmatrix} 1 \\ 1 \\ \end{pmatrix} = \lambda_1 \begin{pmatrix}3 \\ 4 \end{pmatrix} +\lambda_2 \begin{pmatrix} -1 \\ 0 \\ \end{pmatrix} +\lambda_3 \begin{pmatrix} 2 \\ 1 \\ \end{pmatrix}$

$\{(\frac{1}{4},\frac{-1}{4},0)\}$

infinitely many solutions

$\{\}$

$\{(0,1,1)\}$

Question 3

For what value of $a$ are these vectors linearly dependent ?
$\Bigg\{ \begin{pmatrix} -1 \\ 2 \\ 2 \\ \end{pmatrix} , \begin{pmatrix} -2 \\ -2 \\ -2 \\ \end{pmatrix} , \begin{pmatrix} 0 \\ -2 \\ a \\ \end{pmatrix} \Bigg\}$

$0$

infinitely many values

$-2$

$3$

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