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 .

If the vectors in problem above are linearly dependent, write v1 as linear combination of two other vectors.

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

To find the linearly dependent we will use matrix.

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

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 [tex]R^4=4[/tex]

Rank[tex]\neq dim[/tex]

Therefore, it is linearly dependent .

[tex]v_1=(2,-1,0,3)=-\frac{1}{3}(1,2,5,-1)+\frac{1}{3}(7,-1,5,8)[/tex]

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 . If the vectors in problem above are linearly dependent, write v1 as linear combination of two other vectors.

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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 .

If the vectors in problem above are linearly dependent, write v1 as linear combination of two other vectors.