How Do Reflections and Subspaces Challenge Linear Algebra Understanding?

[nicknaq]

Hey everyone! I'd love if you could help me on these couple problems.

Homework Statement

I have attatched a PDF file (don't worry, it will only take you an instant to download - it's just a single page). I'm having problems with numbers 2 and 4.
Number two is about reflections about a line and composition of transformations.
Number four is about subspaces and linear independence.

Homework Equations

Hmm.. I could be wrong, but I don't think 'equations' are needed here. I think it's more applying concepts. Concepts that I am unfortunately not grasping.

The Attempt at a Solution

I honestly have no idea! Sorry. I'm sure you can get me started though. Thanks

 

[Outlined]

2a: find the reflection of the vectors (1, 0) and (0, 1)
2b: watch out, they are talking about another reflection with parameter m'.

4a: look up in your textbook for the definition of vector space, there are 3 rules to check.
4b: Ask yourself whether u, v, 2v or w is in the intersection of E and F.are you a maths student yourself?

 

[nicknaq]

2a: find the reflection of the vectors (1, 0) and (0, 1)

Why? And how? Do I simply multiply (1,0) by the matrix given (i.e. cosΘ, sinΘ, sinΘ, -cosΘ)? If so, where does that get me? I'll have a matrix with Θ terms in it.

2b: watch out, they are talking about another reflection with parameter m'.

I'll work on this after I get (a)

4a: look up in your textbook for the definition of vector space, there are 3 rules to check.

The three rules are:
1. It must go through the orgin
2. closed under additon
3. closed under multiplication
Closed under addition and multiplication I kinda understand, but not really.

4b: Ask yourself whether u, v, 2v or w is in the intersection of E and F.

I'll work on this after I get (a)

are you a maths student yourself?

I'm in a math class, so I suppose yes.

 

[Outlined]

Find the reflection of (1, 0) in the line mx. How? No idea, try sin/cos/tan, pythagoras, orthogonal projections, etc.

Closed under addition means that for every x and y in a subspace V holds x + y is in V.

 

[nicknaq]

Thanks Outlined.

If anyone else has any input that would be greatly appreciated!

 

[nicknaq]

I GOT EVERYTHING EXCEPT FOR 2B and the last part of 3b (sliding it over)
can someone help??


(maybe number 4 too as I'm not sure if I'm right)

 

[Outlined]

For 2b just multiply matrices with each other, you know how a matrix for a reflection looks like (you just find out in 2a), as well how a matrix for a rotation looks like (check your book).

For 3b, I remember you can compute volumes by determinant, however you can also find out the projection of d on the abc plane. Remember: volume = area surface * height.