For (c), suppose that e_1 is in W. Then, there would exist N such that

[math] e_1=\sum_{k=1}^N c_k\sigma_k [/math]

But you showed in (b) that the e_n(k) were an orthonormal basis, so you start taking <·|e_N> on the relation and going down you will discover that all the c_k are 0. This is a contradicition so [math] e_1 \notin W [/math]

For (b), notice that e_n is NOT in W for each n, because <e_n|\sigma_n>≠0. Therefore, W perp is 0. Then you are done

>>8015899
What class is this? Senior level linear algebra?

>>8017428
fuck off, this is a great exercise for a first course in proof based linear algebra, so for mathematics sophomores

>>8017424
These questions are actually really simple. However the wording and symbols used are a bit questionable for any class (if the purpose is to teach and not to wow)

>>8017428
This person is mostly autistic and a 20 something year old under achiever who fans the flames of academic elitism despite having really nothing to show for it

>>8017428
>At most, remedial math the summer before university (and a low ranking one at that)

i bet you have a REALLY high IQ
i feel blessed to be graced by your presence on this lovely night, my good sir

>>8017424
B8.

>>8017438
Never said they were difficult, I was just curious (and didn't see the title was linear algebra part 2). When I was in college I had a similarly named linear algebra class that studied this stuff (most of the class was the Jordan Decomposition theorem, invariance, direct sums, normed vector spaces, and inner product spaces).

>>8017438
He probs is a faggatron

>>8017428
b8, nobody is this autistic.

>>8017462
I don't know if I would say it's senior level but it's certainly a proper linear algebra course for math majors. (As opposed to linear algebra for engineers etc.) Most math majors would take it sophomore or junior year, at least at my uni (a highish-tier American research university).

>>8015899
I took an elementary linear algebra class, and vectors were not mentioned even once throughout the class.

The first part of the class was matrix operations and RREF, the second part of the class was eigenvalues and shit.

That was it.

>>8017532
Sounds like you go to a shitty community college

>>8015899
Just learnt isometry between vector spaces on my last LA2 class. What does the W^{inverted T} stand for?

>>8017812
It's the set of vectors that are orthogonal to each vector in W

>>8017532
What is an eigenvalue? Did you define them to be elements of the set [math]A=\ker (\det(A-xI_n))[/math] or something?

>>8017438
>the wording and symbols used are a bit questionable for any class

It's standard wording and notation for a 2nd year linear algebra class.