Question

* Let S = Span {(2,-1, 1), (3, 1, 1), (1, 2, 0)}. (i) Calculate the dimension of S.

Answer 1

The span of 3 vectors can have dimension at most 3, so 9 is certainly not correct.

Check whether the 3 vectors are linearly independent. If they are not, then there is some choice of scalars $c_1,c_2,c_3$ (not all zero) such that

$c_1 (2,-1,1) + c_2 (3,1,1) + c_3 (1,2,0) = (0,0,0)$

which leads to the system of linear equations,

$\begin{cases} 2c_1 + 3c_2 + c_3 = 0 \\ -c_1 + c_2 + 2c_3 = 0 \\ c_1 + c_2 = 0 \end{cases}$

From the third equation, we have $c_1=-c_2$, and substituting this into the second equation gives

$-c_1 + c_2 + 2c_3 = 2c_2 + 2c_3 = 0 \implies c_2 + c_3 = 0 \implies c_2 = -c_3$

and in turn, $c_1=c_3$. Substituting these into the first equation gives

$2c_1 + 3c_2 + c_3 = 2c_3 - 3c_3 + c_3 = 0 \implies 0=0$

which tells us that any value of $c_3$ will work. If $c_3 = t$, then $c_1=t$ and $c_2 = -t$. Therefore the 3 vectors are not linearly independent, so their span cannot have dimension 3.

Repeating the calculations above while taking only 2 of the given vectors at a time, we see that they are pairwise linearly independent, so the span of each pair has dimension 2. This means the span of all 3 vectors taken at once must be 2.