Linear Algebra Hw Help: Subset Independence & Dimension | C(R) Functions"

[bor0000]

Hi please help me out!

1a) Determine whether this subset of C(R) is linearly independent or not; the dimension of the subspace and its basis
S=(e^ax, x*e^ax, x^2*e^ax)
this question seems to be too easy, they seem independent to me for obvious reasons. adn those 3 elements are the basis, and the dimension is then obviously 3. or am i wrong somewhere?

and it asks to show that one can find arbitrarily large finite sets of independent functions in C(R). then i take it x, x^2, x^3, etc are independent, now how do i show/prove this?


thanks!

 

[HallsofIvy]

C(R) is the set of functions continuous on all real numbers.

Yes, S is a set of independent functions. I don't know what your "obvious reasons" are- can you show that the definition of "independent" holds? Yes, since they are three independent "vectors", the subspace they span has dimension 3 and S is itself a basis.,

Given any n, show that the functions y= x⁰= 1, x¹, x², ..., xⁿ are independent. What IS the definition of "independent"?

 

[bor0000]

thanks! I am just starting to do this homework today. I think i know how i'll show this, set it up as some kind of matrix, and try to get it so that each row has more zeroes than the previous one.

 

[HallsofIvy]

Well yes, you could do that- I think that's "overkill".


Suppose a₀x⁰+ a₁x¹+ a₂x²+ ...+ a_nx_n= 0 (for all x). Take x= 0 and you get a₀= 0! Now differentiate both sides of that:
a₁+ a₂x+ a₃x²+ ...+ a_nx_n-1= 0. Let x= 0 again: a₁= 0. Continue!

 

[matt grime]

Or you could use the fact that a degree n poly only has at most n roots.

 

[bor0000]

thanks! i don't understand how i can use the 'derivatives' or 'roots' making it independent. anyway i did those problems by setting up a matrix like 1*a0+0*a1+0*a2 for row1, etc. it was simple.
what about #4 here
http://www.math.mcgill.ca/schmidt/247w05/ass2.pdf
sorry I'm posting my whole assignment, i don't intend to copy all of it, i can also go to professors' office in the morning or helpdesk in the afternoon, but i am busy studying chem homework now, and so don't know if i'll make it.
so in #4, i set up a matrix and labelled it horizontally as the normal basis, i.e. 1,x,x^2..(i don't think i need to prove that its a basis for p(n)), label it V, and vertically the basis that was given in question, label it U. Then i would need to prove that every element in U can be expressed uniquely as a combination of V's. i.e. column 1 will be 1,0...,0, column2 will be -a,1...0, and so i need to prove that it will have a triangular form like this. I have no way to formally prove this, but i wonder if its enough that it just seems to me intuitively that they have the same power j, and the other elements have powers less than j, and so the elements in the column below j will be 0, but whatever they are above j doesn't matter, since they will be something.
So please tell me if this is enough, or if not, any other ideas to do this?
And for the second part where they ask for the coefficient of p(x), does it mean they want the coefficient of the largest term, i.e. 2x^3+1 would have a coefficient 2? Then i used the taylor expansion and got D^k(a)/k!

And in #6a) what is reflection?? I looked in my notes and i am not sure about this, but i think it says that if x has 2 vectors, in this case v1=(1,0,1) and v2=(0,1,-2), then find the vector that is orthogonal to them, which is 1,-2,-1 , and to get the reflection just right these 3 vectors as columns except that orthogonal one is multiplied by -1. I drew a parallipeped to picture this, and i don't understand why this would mean it's a reflection.

And in 6b) to get P i also wrote the 3 vectors as columns, but set the orthogonal vector to 0,0,0 but then for Q do i do the same but set one of the other vectors to 0? if so which one and why?

thanks!

 

[matt grime]

Go ask your professor to explain what zero is.

 

[matt grime]

Ok, let me try to be more helpful

You want to show that $$\sum_{r=0}^{n} a_r x^r$$ is the ZERO FUNCTION, that is it is zero for all values of x. A degree n polynomial, which that is, has at most n roots for n greater than 1 (I am assuming a_n is not zero, which we may do). So if it were the zero function (ie all values of x are roots) then you have a cotnradiction.

 

[fourier jr]

find the determinant of: $$\left( \begin{array}{ccc}f & g & h \\f' & g' & h' f" & g" & h"\end{array} \right)$$, where f g & h are the functions you want to show are linearly independent, & see if it's zero anywhere.

edit: wtf why doesn't my tex ever work? it always says 'page cannot be displayed'

 

[bor0000]

ok thanks!

 

[mathwonk]

one satndard trick to show functiuons are independent is to assume a dewpendency relation and then differentiate it to get more, eventually one whioch is impossible.

e.g. to show e^ x and e^(2x) ibdependent, assume that ae^x + b e^(2x) = o (the zero function.

then also by differentiating, ae^x + 2be^(2x) = 0, so if x = 0, i get a + 2b = 0.

but in the original equation, x=0 gives a+b = 0, so b = 0. but then i have ae^x = 0, so putting x=0 again gives me a = 0.