Question

Which of the following sets are linearly independent?

A. { (1, 0) , (1, 1) , (1-1) } in R2

B. { (1, 1, 1), (1,-1, 1), (-1, 1, 1) } in R3

C. { 1 + x, x, 2 + 3x } in P2

D. { [1 1 , [1 1

        0 0] , 0 1] } in M22

Select from the following:

1. Only A and C.

2. Only B.

3. Only D.

4. Only B and D.

5. None of the above.

Thank you

Answer 1

Linear dependence indicates whether we can write the given vectors as a combination of any other vectors present in the set or not.

Based on this,

Let's take option A

2(1,0) -1(1,1) = (1,-1) So, Not possible

Consider Option B

These vectors, are linearly independent as we can not write them in kV1+jV2 = V3 form

Consider Option C

2(1+x) +x = 3x+2

So it's linearly dependent and ruled out

Consider Option D

It is also not possible to represent linear dependency there

So, option B and D are linearly independent