Linear algebra

Question

How to determine whether the vectors u:(1,-1,4) , v:(-2,1,3) , w:(4,-3,5) span R^3     ???

Answer 1

To span R3, an arbitrary vector, let's call it b=(b1,b2,b3), must be a linear combination of u v and w.

that means

b= K1u + k2v + k3w where k1, k2,k3 are constants.

 

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

 

So b1 = k1 -2k2+4k3

     b2 = -k1 + k2 -3k3

     b3 = 4k1 + 3k2 + 5k3

 

This leads to the matrix

    /  1 -2 4 /

    / -1 1 -3 /

    / 4 3 5   /

 

If this matrix has an inverse, it means that the system above can be solved and therefore these 3 vectors span R3.

To find this out, you just to see if the determinant is not zero.

 

The determinant is 1(5--9) - -2(-5 - -12) + 4(-3 - 4) = 14 +14-28 = 0. Since the determinant of this matrix is 0, this matrix is not invertible and the system cannot be solved, therefore these vectors do not span R3.

 

Hope this helped.