Linear Algebra question

Question

check that B ={ [-1/2,√3/2,0], [√3/2,1/2,0], [0,1,-1] } is a basic for ℝ³. Then, determine whether it is orthogonal, orthonormal or neither.

Answer 1

We have to check if this 3 vectors linearly independent, if determinant with columns- coordinate of this vectors not equel 0:

I -1/2 √3/2 0 I

I √3/2 1/2 1 I = 1/4 + 3/4 = 1 ≠ 0; we have bases

I 0 0 -1 I

Now we check if this vectors orthogonal to each other. WE take dot-products:

<-1/2,√3/2,,0>•<√3/2,1/2,0> = -√3/4 + √3/4 = 0

<-1/2,√3/2,,0>•<√0,1,-1> = √3/2 ≠ 0

So, this bases not orthogonal

This bases not orthonormal because for example absolute value of vector <0, 1, 1> is √2 ≠ 1