Check if vectors u 213 v 101 w 415 are linearly independe

Check if vectors u = [2,-1,3], v = [1,0,1], w = [4,-1,5], are linearly independent: if not, find a nontrivial linear combination of them that?s equal to 0.

Solution

Here given vectors u=(2,-1,3), v= (1,0,1) and w= (4,-1,5) are linearly dependent because their determinant is zeo because

| 2 -1 3 |

| 1 0 1 | = 2[0(5) - (-1)(1) ] - (-1)[(1)(5)-(1)(4)] +3[(-1)(1)-0(4)] = 2(0+1)+1(5-4)+3(-1-0) =2+1-3=0

|4 -1 5 |

and as they are linearly dependent, one vector can be represented as the linear combination of other two or

(2i-j+3k)=x(i+k)+y(4i-j+5k)

=(x+4y)i -yj +(x+5y)k

or on comparing both sides,

-y=-1 or y=1

and x+4y=2

or x+4=2

or x=2-4= -2

So required non trivial linear combination is

1(2i-j+3k)-2(i+0j+k)+1(4i-j+5k)=0