my wife needs help with a problem for her linear algebra class, with an explanation on how to work through it if anyone can help.

here is the problem

construct an orthonormal basis for the subspace of R^3 spanned by the following vectors.

(a) (1, 0, 2), (-1, 0, 1)
(b) (1, -1, 1), (1, 2, -1)


thanks for ay help in advance, i would help her myself, but everything i learned in college i forgot before i walked down to pick up my diploma.

 

She should have learned the Gram-Schmidt algorithm for constructing an orthogonal basis, from which an orthonormal basis is a trivial step.

Call the given vectors V1, V2, . . ., Vn. (In this case, it's only V1 and V2 for each problem.)

Call the vectors in an orthogonal basis U1, U2, . . ., Un.

Choose U1 = V1.

To get U2, start with V2 and subtract the projection of V2 onto U1.

To get U3, start with V3 and subtract both the projection of V3 onto U1 and the projection of V3 onto U2. In general, for Uk, start with Vk and subtract all of the individual projections of Vk onto U1, U2, . . ., U(k-1).

Finally, to get an orthonormal basis, divide each of the vectors in the orthogonal basis by its length.

(I'll be teaching linear algebra and differential equations this Spring.)

 

Bill,

I'm a few classes short of a math major and yet I don't remember what you're talking about. I did it, and knew it, but it's been years since I used it. Scary, huh? Every time I think of how much I've forgotten, I wince.

Jon

 

A couple of days ago my wife and I were talking about the possibility of me going back to school to finish up my PhD. I was saying that it will take me at least a year of refresher courses simply to recall all that I've forgotten since graduate school.

 

i know exactly what you both mean, i got my math minor only 3 years ago, and even had an "A" in linear, but i have no clue how to help my wife with these problems, they just look greek to me. i know that at one time i could have done it no problem, but not any more.


on another note, magician, i showed my wife your insight and she said that is basically what she has been doing, although she said that she may have messed up at the last step where you said to divide each of the vectors in the orthogonal basis by its length. so she is going to give it another shot.

thanks

 

(a)

U1 = (-1, 0, 1) and U2 = (-6/5, 0, 3/5) by G-S.

||U1|| = sqrt(2), ||U2|| = sqrt(9/5).

An orthonormal basis is (-1/sqrt(2), 0, 1/sqrt(2)), ((-6/5)/sqrt(9/5), 0, (3/5)/sqrt(9/5)).

(b)

U1 = (1, -1, 1), U2 = (5/3, 4/3, -1/3) by G-S.

||U1|| = sqrt(3), ||U2|| = sqrt(42/9).

An orthonormal basis is (1/sqrt(3), -1/sqrt(3), 1/sqrt(3)), ((5/3)/sqrt(42/9), (4/3)/sqrt(42/9), (-1/3)/sqrt(42/9)).

 

she got it figured out, thanks for your help.

turns out that there must have been a misprint in her answer key

 

Always a pleasure.