Question

Determine if the vectors v1 = (2, −1, 0, 3), v2 = (1, 2, 5, −1), v3 = (7, −1, 5, 8) are linearly independent vectors in R4 .

Answer 1

Answer with Step-by-step explanation:

We are given that three vectors

$v_1=(2,-1,0,3), v_2=(1,2,5,-1),v_3=(7,-1,5,8)$

We have to determine the given vectors are linearly independent in $R^4$ and write $v_1$ as linear combination of other two vectors if the vectors are dependent.

To find the linearly dependent we will use matrix.

$\left[\begin{array}{cccc}2&-1&0&3\\1&2&5&-1\\7&-1&5&8\end{array}\right]$

If m=Number of rows, n=Number of columns then,

Rank of matrix=min(m,n)

Rank of matrix=min(3,4)

Rank of matrix=3

Dimension of $R^4=4$

Rank$\neq dim$

Therefore, it is linearly  dependent .

$v_1=(2,-1,0,3)=-\frac{1}{3}(1,2,5,-1)+\frac{1}{3}(7,-1,5,8)$