Linear algebra vectors form basis for R^3?

Question

How do I solve the following problems? (I don't want just the answer--I already have those. I want to know how to do these for the test) : 
Determine whether the following set of vectors form a basis for R^3. Justify your answer.
a. {v1 = (1,1,0)^T, v2=(1,0,1)^T, v3=(0,1,-1)^T}
b. {v1=(3,-2,1)^T, v2=(2,3,1)^T, v3=(2,1,-3)^T}

Thanks!

EM · Answer

A set of 3 vectors would form a basis for R3 when each basis vector is not a linear combination of the other two. We could check for linear dependence of these vectors by solving a(v1) + b(v2) + c(v3) = 0 for non-zero a, b, c. Or we could just recognize that if the determinant formed by the rows or columns of the proposed basis is zero, then the vectors are linearly dependent.

a.
| 1 1 0 |
| 1 0 1 | =
| 0 1 -1 |

(expansion by minors across first row):
(1)| 0 1 | - (1)| 1 1 | + (0)| 1 0 | =
... | 1 -1| ..... | 0 -1| ..... | 0 1 |

(1)(-1) - (1)(-1) + 0 =

0 (so the vector set does not form a basis for R^3)

b.
| 3 -2 1 |
| 2 3 1 | =
| 2 1 -3 |

(expansion by minors across first row):
(3)| 3 1 | - (-2)| 2 1 | + (1)| 2 3 | =
... | 1 -3| ..... | 2 -3 | ..... | 2 1 |

(3)(-10) - (-2)(-8) + (1)(-4) =

-50 (so this set does form a basis for R^3)

Linear algebra vectors form basis for R^3?

1 Answer