Define Matrix A w/ Orthogonal Vectors | R3

[dlevanchuk]

Homework Statement

Let S = { u1, u2, u3} be an orthogonal set of nonzero vectors in R3. Define (3 x 3) matrix A by A = [u1, u2, u3]. Show that A is nonsingular and A'A (' is transpose) is a diagonal matrix. Calculate the diagonal matrix using the given orthogonal vectors: u1 = [1 1 1]'; u2 = [-1 0 1]'; u3 = [-1 2 -1]'

Homework Equations

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The Attempt at a Solution

I think my IQ drops by the minute, thereof a stupid question... what does it mean to "define" the matrix? Do I just create the matrix from the given set of vectors??

 

[radou]

I think my IQ drops by the minute, thereof a stupid question... what does it mean to "define" the matrix? Do I just create the matrix from the given set of vectors??

Yes, exactly.

Now, what is the relation between orthogonality and linear independence of vectors? Further on, how does this relate to the matrix rank? And how does the rank relate to regularity/non-singularity?

 

[dlevanchuk]

could i just say that since the vectors are orthogonal, that means they are linearly independent, and that makes the matrix non singular.. sounds pretty logical to me :)