Lineer algebra - maximum number of linearly independent vectors

Question

Question: Decide on the maximum number of linearly independent vectors among a given a set of vectors. $$ A = (1,-1,0,0) \quad B = (1,0,-1,0) \quad C = (1,0,0,-1) \\ D = (0,1,-1,0) \quad E = (0,1,0,-1) \quad F = (0,0,1,-1) $$

I don't understand that question. how can I solve this question?

edit: Actually in that question, I though like that, how can write that vectors with another two vectors and so how many vectors there are like that?? is this true. can anyone explain

I mean,

A-B = (0,1,-1,0)

so, A-B = C I can write C instead of A and B.. It is my understanding.

Answer 1

If your set consists of vectors $\{u_1,u_2,\dots u_k\}$, it may not be a linearly independent set but you can always take a subset that is linearly independent (usually we want the maximum number of these). For example if my set consists of $u_1=(1,1), u_2=(0,1),u_3=(2,3)$, then a set with maximum number of linearly independent vectors, that I can take from this set is $\{(1,1),(0,1)\}$ since $u_3$ is a linear combination of the first two. Equally it would be fine to take any two here, since any pair out of this set will be linearly independent.

As pointed out in the comments, you probably were given a procedure to follow. If unsure of why and how that works, you can ask.

EDIT:

Putting your vectors into a matrix, as columns, and row reducing will give a rank 3 matrix, which means we have three linearly independent vectors. The first three columns consist of elementary vectors and the remaining columns contain coefficients of linear dependence for the corresponding vectors. Columns 4 is $(-1,1,0,0)$ which tells me that $D=-A+B$

Answer 2

E.g. if $\{ A, B, C \}$ are linear independent, then the equation $$ c_1 A + c_2 B + c_3 C = 0 $$ has only the trivial solution $c = (c_1, c_2, c_3) = (0, 0, 0) = 0$.

There are different methods to determine this. E.g. you can formulate this as system of linear equations $$ (A^\top B^\top C^\top) c = M c = 0 $$ and use Gauss elimination to find out the rank of $M$, where $\top$ means matrix transposition (here turning row vectors into column vectors).

You can think in advance about how many vectors can be linear independent at most.

Choosing the right subset is bit trial and error.