Question

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..

Answer 1

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

Answer 2

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

Answer 3

I don't know what to do after..

Answer 4

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...

Answer 5

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

Answer 6

welcome :)