Question

S = { (1,2,3) , (4,5,6) , (11,16,21) }
Is the span of the vector group S = R3

Answer 1

I've applied the gaussian elimination method and got this:

[ 2 5 16 ]
[ 0 6 12 ]
[ 0 0 -9 ]

Answer 2

Is it safe to say 2, 6 and -9 are the leading coefficients and therefore S spans R3?

Answer 3

You will have to notice that the vectors you started with are linearly independent, so they span a vector subspace of \( R^3\) of dimension 3. So it must be the whole of \( R^3\)

Answer 4

@Ankh to establish that the vectors are linearly independent, you need to show that the linear combination of the vectors sums to the zero vector for non-trivial coefficients \(c_1,c_2,c_3\) in
\[c_1\vec{v}_1+c_2\vec{v}_2+c_3\vec{v}_3=\vec{0}\]