Do the vectors V1201V2003V3110 form a basis for R3Justify Wh

Do the vectors V1=(2,0,1),V2=(0,0,3),V3=(1,1,0) form a basis for R^3?Justify .What is the span of V1,V2,V3?

Solution

The vectors form a basis for R^3 if they are linearly independent vectors i.e. the only solution possible

aV1 + bv2 + cv3 = 0 is (a=b=c=0)

substituting the vectors we get

a(2,0,1) + b(0,0,3) + c(1,1,0) = 0

2a + c = 0

c = 0

a + 3b = 0

c is equal to 0, that implies 2a + c = 0 => a =0

0 + 3b = 0

b = 0

Hence the only solution possible for this set is (a=b=c) is 0, hence the span the R^3 space since they form the basis

We can also prove the other thing by finding out the determinant of the matrix and proving that it is not equal to zer0

Determinant = 2 ( 0*0 -1 * 3) + 0(1-0) + 1(0-0) = -6

Since determinant is not zero, hence vectors are linearly independent and hence they form the basis of R^3

2	0	1
0	0	1
1	3	0