Spanning Sets in R3: Determine Vectors

[TaliskerBA]

Homework Statement

Determine whether the given vectors span R3:
v1=(3,1,4) v2=(2,-3,5) v3=(5,-2,9) v4=(1,4,-1)

Homework Equations

I need to show that an arbitrary point in R3 can be written as:

(b1,b2,b3)=k1(3,1,4)+k2(2,-3,5)+k3(5,-2,9)+k4(1,4,-1)

The Attempt at a Solution

I know that when you have 3 different vectors and have to work out if they span R3 you can write the coefficient matrix and find out if the determinant of that matrix is equal to zero or not.

Therefore I know that:

3k1 + 2k2 + 5k3 + k4 = b1
k1 + (-3)k2 + (-2)k3 + 4k4 = b2
4k1 + 5k2 + 9k3 + (-1)k4 = b3

Since this augmented matrix is 3x4 I can't use the determinant method I used before (unless I use a long-winded method of working out if 3 vectors span at a time).

I apologise if this is quite basic I am only just learning about vector spaces!

 

[Mark44]

If you're familiar with the term augmented matrix, then maybe you're familiar with the idea of reducing a matrix using row operations. If so, row reduce the augmented matrix. If you end up with three nonzero rows, the vectors span R³. If you end up with fewer than three nonzero rows, the vectors don't span R³.

 

[TaliskerBA]

Yeah that works, thanks!