Which of the following vectors are in span { ~v1, ~v2, ~v3}

Suppose that ~v1 = (2, 1, 0, 3), ~v2 = (3, −1, 5, 2), and ~v3 = (−1, 0, 2, 1). Which of the following vectors are in span { ~v1, ~v2, ~v3}?

i. (2, 3, −7, 3)

Is this not in span? I got that it was in span, but a website I am using to double-check my answers says that it is NOT in span. Please help!

Solution 1:

You are right, the given vector lies in the span of the three vectors $v_1, v_2, v_3$. In general we have a few options how to check if a given vector $w$ lies in span$(v_1, \dots, v_n)$.

One option is to solve the system of linear equations $$x_1v_1 + \dots + x_nv_n=w$$ which can be written in matrix form: $$(v_1 |\dots |v_n)x=w$$ where the matrix on the left has the $v_i$ as columns. (In our case this system is solvable and gives us that $2v_1-v_2-v_3=(2,3,-7,3)$)

A second option is to calculate the ranks of the following two matrices: $$(v_1|\dots|v_n)$$ and $$(v_1|\dots|v_n|w)$$ If the rank of both matrices is equal, $w$ lies in span$(v_1, \dots, v_n)$ - if the rank of the second matrix is bigger than the rank of the first matrix that is not the case. In our case we would get that the rank of both matrices is $3$.