Can Linear Algebra Proofs Be Mastered with the Right Strategies?

[Übermensch]

Homework Statement

I'm currently in first year linear algebra... I'm doing quite well, there's just one area of trouble-- proofs. For example:

Suppose u.v = u.w, does it follow that v = w? Prove your generalization.

Prove that u is orthogonal to v - proju(v) for all vectors u and v in R^n where u != 0.

Prove that (u + v) . (u - v) = ||u||^2 - ||v||^2 for all vectors u and v in R^n.

There are about 20 questions in my current assignment in this format. I haven't been able to answer one of them to my satisfaction, whereas I currently have all non-proof questions correct.

Homework Equations

This is the problem. It could be anything. I have hundreds of equations with these variables in them... But in a test situation, I couldn't possibly try all possible equations and see if they yield anything useful.

The Attempt at a Solution

This is also a problem. I haven't the slightest clue where to start. If I had a beginning point, or a way to find a beginning point, I might actually be able to do these questions. :)

Edit: perhaps there are some good websites that may have linear algebra proofs and other equations to practice with? There are so many sites out there, and the 20 or so I looked at today didn't have much material that I didn't already know... But there has got to be a good one somewhere.

 

[cristo]

1. What is the definition of the dot product?
2. I don't know what you mean by proju(v).
3. Try using the distributivity of the dot product to expand the LHS, then simplify using commutivity.

 

[Übermensch]

Alright, I'll try those out tomorrow (I'm not at home right now.)

Do you have any general tips for what I should do for proofs? Are there any general suggestions about solving them that I'm missing?

As for #2, it's just "projection of v onto u.

Thanks! :)

 

[cristo]

Alright, I'll try those out tomorrow (I'm not at home right now.)

Do you have any general tips for what I should do for proofs? Are there any general suggestions about solving them that I'm missing?

Erm.. not particularly. I would just advise to always write down the definitions of things you are trying to prove, and work from there. I guess someone else will be able to give more tips.

As for #2, it's just "projection of v onto u.

Ok, so do you have a definition for this projection operator? Use the definition then dot u with v-proj u(v) and see if it equals zero.

Thanks! :)

You're welcome!