1. ## Find the Basis

for the each of the following vector spaces V, find the Basis and give the dimension:
a) V = [ 3x3 real matrices]
b) V= smooth real-valued functions on [0,1] satisfying y'''-y' = 0]
c) V= [real valued functions on a set X containing n points]
d) V= [complex 2x2 matrices A for which Atr =A (where A is the matrix of complex conjugates) ]

Attempt:

a) standard basis with dim 9
b)I actually don't know where to start on this one!
c) a wild guess that dim v = n but again i'm struggling to think of the entries for the basis
d) unsure but I have had a guess and I think it is dim 3 with these entries
a+ib c+id a-ib c-id a+ib c-id
c+id a+ib c-id a-ib c-id a+ib

I appreciate any help I recieve. Thank you.

2. ## Not sure, but...

You can make bases out of the following:

1. 9 matrices, each 3x3, each with a 1 in one of the matrix element slots and zeroes otherwise. I assume this is what you mean by the "standard" set.

2. Either y'=0, or y''=y with y' not equal to 0. If y'=0, a basis is the function 1 spanning the set of y= constant functions. If y''=y, you have linear combinations of exp(x) and exp(-x), so a basis could be these 2 functions.

3. Define a function to be a set of N points, with the m'th point being the value of the function on the m'th point in the set. Then a basis would be the standard N-component unit vectors (1 in one position, 0 otherwise).

4. Are you working over C or over R? The 2x2 complex matrices have 8 degrees of freedom if over R, and 4 if over C - without your additional condition. The condition A transpose = A complex conjugate imposes the condition that the diagonal elements are real, and the off diagonal elements are related - just write out both of the matrices like you did in the problem and see what the condition implies in terms of the matrix elements.

3. Originally Posted by qmech
You can make bases out of the following:

1. 9 matrices, each 3x3, each with a 1 in one of the matrix element slots and zeroes otherwise. I assume this is what you mean by the "standard" set.

2. Either y'=0, or y''=y with y' not equal to 0. If y'=0, a basis is the function 1 spanning the set of y= constant functions. If y''=y, you have linear combinations of exp(x) and exp(-x), so a basis could be these 2 functions.
So for the entire problem, a basis consists of $\{1, e^{x}, e^{-x}\}$.

3. Define a function to be a set of N points, with the m'th point being the value of the function on the m'th point in the set. Then a basis would be the standard N-component unit vectors (1 in one position, 0 otherwise).

4. Are you working over C or over R? The 2x2 complex matrices have 8 degrees of freedom if over R, and 4 if over C - without your additional condition. The condition A transpose = A complex conjugate imposes the condition that the diagonal elements are real, and the off diagonal elements are related - just write out both of the matrices like you did in the problem and see what the condition implies in terms of the matrix elements.

4. Quote:
3. Define a function to be a set of N points, with the m'th point being the value of the function on the m'th point in the set. Then a basis would be the standard N-component unit vectors (1 in one position, 0 otherwise).

I am unsure with regards to 3.
does this mean dim V = n?

Thank you for you help above, I've got 1 and 2 now fully understood, working on 4 now but I think I understand it from above.