Linear Dependence of Vectors: a, b, c

[theBEAST]

Homework Statement

Are the vectors a = [1 -1 0 1], b = [1 0 0 1] and c =
[0 -1 0 1] linearly independent?

The Attempt at a Solution

I am mainly confused about whether or not I should have my matrix in row or column form to solve this:

r 1 -1 0 1
s 1 0 0 1
t 0 -1 0 1

or
r s t
1 1 0
-1 0 -1
0 0 0
1 1 1

 

[HallsofIvy]

The standard method is to write the matrix having the vectors as columns. However, that is only a convention. Doing it the other way should give the same result- the vectors will be independent, doing this the first way, if you do NOT get a row of all 0s. Doing it writing the vectors as rows, the vectors will be independent if you do NOT get a column of all 0s.

However, if you have trouble remembering this, it appears you are trying to memorize a method that is, to you, arbitrary. Personally, I prefer to use the definition of "linear independent" which you should have learned anyway. A set of vectors is "linearly independent" if and only if the only linear combination that is equal to the 0 vector has all coefficients equal to 0. That is, in this case, if a<1, -1, 0, 1>+ b<1, 0, 0, 1>+ c<0, -1, 0, 1>= <0, 0, 0, 0> then we must have a= b= c= 0. Is that true?

Of course, that gives <a+ b, -a- c, 0, a+ b+ c>= <0, 0, 0, 0> or the three equations a+ b= 0, -a-c= 0, a+b+ c= 0. From the first b= -a and from the second, c= -a. Put those into the third equation and solve for a.