linear algebra: subspaces Which of the following vectors in R4(1xn matrix) are linear combinations of: v1 = [1 2 1 0] v2 = [4 1 -2 3] v3 = [1 2 6 -5] v4 = [-2 3 -1 2]? (a) [3 6 3 0] (b) [1 0 0 0] (c) [3 6 -2 5] (d) [0 0 0 1] answer: (a) and (c) how do you solve this question using linear combinations of 1xn matrices? I can do questions involving nx1 matrices but this one confuses me..

The way I see it is top make a matrix of all the vectors as rows\[\left[\begin{matrix}1&2&1&0\\4&1&-2&3\\1&2&6&-5\\-2&3&-1&2\end{matrix}\right]\]now row reduce as much as possible from there you should have simpler vectors to work with, and any vector you can form as a linear combination of those vectors should also be a possible linear combination of the original vectors

I row reduced it and I got: \[\left[\begin{matrix}1 & 2 & 1 & 0\\ 0 & 1 & 0 & 3/7 \\ 0 & 0 & 1 & -1 \\ 0 &0 &0 &0\end{matrix}\right]\]

I don't know what to do after..

we can form the first vector (a) from this; it's just 3 times the first row by inspection, 3 times the first row plus -5 times the third row is the other vector (c) the other vectors cannot be formed if there's another way to do this I'm not familiar with it, I just tried to reason it on the fly *many typos...

Ohh, I see. Thanks again TuringTest! that cleared it up :)

welcome :)

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