Question

Check whether the set of vectors v1=(1,1,0,1), v2=(1,0,2,1), v3=(-1,1,-3,-2) €R^4 are linearly independent. If they are dependent, find a1,a2 and a3 €R ,not all zero, such that a1v1+a2v2+a3v3=0.

Answer 1

Answer on Question #85744 – Math – Algebra

Question

Check whether the set of vectors $v1 = (1, 1, 0, 1)$, $v2 = (1, 0, 2, 1)$, $v3 = (-1, 1, -3, -2)$ ∈ $\mathbb{R}^4$ are linearly independent. If they are dependent, find $a1$, $a2$ and $a3$ ∈ $\mathbb{R}$, not all zero, such that $a1v1 + a2v2 + a3v3 = 0$.

Solution

Determine if the set of vectors is linearly independent solving the following equation:

$$a1 \begin{bmatrix} 1 \\ 1 \\ 0 \\ 1 \end{bmatrix} + a2 \begin{bmatrix} 1 \\ 0 \\ 2 \\ 1 \end{bmatrix} + a3 \begin{bmatrix} -1 \\ 1 \\ -3 \\ -2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$$

or what is the same, the set of equations:

$$\begin{cases} a1 + a2 - a3 = 0 & (1) \\ a1 + a3 = 0 & (2) \\ 2a2 - 3a3 = 0 & (3) \\ a1 + a2 - 2a3 = 0 & (4) \end{cases}$$

Subtract (4) from (1):

$$\begin{cases} a1 + a2 - a3 = 0 \\ a1 + a3 = 0 \\ 2a2 - 3a3 = 0 \\ a3 = 0 \end{cases}$$ $$\begin{cases} a1 + a2 = 0 \\ a1 = 0 \\ 2a2 = 0 \\ a3 = 0 \end{cases}$$ $$\begin{cases} a2 = 0 \\ a1 = 0 \\ a2 = 0 \\ a3 = 0 \end{cases}$$

As $(a1, a2, a3) = (0, 0, 0)$ is the only solution of the system then the given vectors are linearly independent.

Answer: the vectors are linearly independent.

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