Determine whether the vectors in the set S span the vector s

Determine whether the vectors in the set S span the vector space V.

V=R²; S = {[0, 0], [1,1]}

The answer is: \"The given vectors do not span R², although they span the one-dimensional subspace {k[1, 1] | k E R}\"

Can you please explain what this answer means?

Also, the solution says that a=0 and b=c₁+c₂. When I am doing this problem, I keep getting that a=c₂ and b=c₂. How am I doing this wrong? Thanks!

Solution

c1(0 , 0) + c2(1 , 1) = (a , b)

(0 , 0) + (c2 , c2) = (a ,b)

(c2 , c2) = (a , b)

a = c2 and b = c2

So, as can be seen the only kinds of vectors that can be spanned by (0,0) and (1,1) is of the form (c2,c2)
So, they are all vectors where the x-value = y-value.....

So, <1,1> * k ; where k can be any real value..

In simple terms, we can generalize such a vector as {k<1 , 1> ; k E R}

Hence proved!