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Suppose B is a vector and equal to (a,b,c,d) and also suppose that B lies in the span of (1,0,2,1), (-2,3,-1,1) and (2,-2,1,-1). What conditions must a,b,c and d satisfy?

This is something that i have no idea how to approach. i know the answer is a - c + d = 0 but i don't know how to get to this answer.

can anyone point me in the right direction?

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  • $\begingroup$ Hint: Consider the reduced row echelon form of $\begin{bmatrix}1&0&2&1\\-2&3&-1&1\\2&-2&1&-1\end{bmatrix}$ $\endgroup$
    – JMoravitz
    Oct 13, 2017 at 22:37

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We are looking for the coefficients $x,y,z,w$ of linear equation(s) $$xa+yb+zc+wd=0\quad(1)$$ which determines $U$ by $a,b,c,d$.

Now, we know the 3 given vectors satisfy $(1)$, thus we already have $3$ equations for $x,y,z,w$ that we can solve: $$\pmatrix{1&0&2&1\\-2&3&-1&1\\2&-2&1&-1}\cdot\pmatrix{x\\y\\z\\w}=\pmatrix{0\\0\\0}$$

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