Struggling with Linear Algebra? Need more practice problems and answers?

[MtHaleyGirl]

I don't know if it is the subj or my prof but I am having a hard time grasping Linear Algebra. We are working on vector spaces - subspaces, basis & dimension, nullspace...etc. The text we have is LINEAR ALGEBRA WITH APPLICATIONS 7E by Steven J. Leon. I don't think there are enough problems for me to work through in the text. Does anyone know a website or online text that has good problems and answers so I can get some practice in this stuff! Answers are critical and any thing that includes steps on how that answer was obtained would be awsome... ? Or maybe there is somebody out there just itching to explain subspace & basis <<< mostly basis is where I am stuck right now >>> Thanks

 

[mathwonk]

the key idea is linear combinations. they are a way to, express an infinite number of things usaing only a finite number of basic ones.


e.g. every polynomial of degree 2 or less can be expressed as a sum of multiples of (i.e. as a linear combination of) the absic monomials, 1,X,X^2. Hence the vector space of all polynomials of degree 2 or less has dimesion at most 3 (the number of elements in our set 1,X,X^2).

the set 1,X,X^2 which can be used to express everything contain s a basis. To see if it is itself a basis, you must check whether any polynomial of degree 2 or less can be expressed using these guys, in more than one way.

i.e. suppose a + bX + cX^2 = d + eX + fX^2. Can you be sure that a=d, b=e, and c = f? if so, then there is only one way to use the basic monomials 1,X,X^2 to express any degree 2 polynomial, and so they are a "basis". I.e. they not only "span" everything, but they are also independent, hence a basis.

(My use of the word "basic" earlier was merely informal, and had nothing to do with the technical meaning of "basis".)


the functions 1, cos^2 and sin^2, are not independent, since 1 = cos^2 + sin^2. so there is more than one way to express the function 1 using them.

first master linear combinations, then an unnderstanding of bases will be added unto you.


i.e. given a collection {v1,...vn}, the set V f all things that are linear combinations of these, i.e. all things of form a1v1+...anvn, is the subspace spanned by the set v1,...,vn.

And if each thing in that subspace V has exactly one expression as a linear combination of the set v1,...vn, then that set is a basis of that subspace.

in fact v1,...,vn is a basis of V if and only if no one of the vj can be expressed as a linear combination of the other v's.


so all the concepts you are asking about are defiend in terms of linear combinations.

 

[matt grime]

degree 2 polys have dimension 3.

 

[mathwonk]

thanks matt, i went back and changed the 2 to a 3 where i noticed it.

 

[HallsofIvy]

Now go back and change "dimension at most 3" to "dimension 3". You are, after all, talking about the vector space of all degree two polynomials.,

 

[MtHaleyGirl]

thank you, mathwonk, for taking the time to explain that. It helps.

 

[mathwonk]

thanks for the remark halls, but i don't want to change that statement, since at the time all i had done was prove my set was spanning, and not yet that it was a basis. so my argument showed not that the dimension was equal to 3, but only that it was at most 3.

I never did finish the argument that it was a basis, so I was never entitled to say the dimension equalled 3. I left that as an exercise for the reader.

but of course you are right, once one does prove that the set 1,X,X^2, is also independent, it follows that the space of polynomials of degree at most 2, is actually 3.

 

[mathwonk]

one also needs to be careful about the definition of a polynomial. although with the standard definition, that two polynomials are equal if and only if their coefficients are all equal, the dimension of the space of polynomials of degree at most d is indeed d+1, some students confuse the concept of formal polynomial with that of polynomial function.


A polynomial function is the image of a polynomial under the map from polynomials to functions. Over the field Z/2 e.g. X^2 = X, so as polynomial functions X^2 and X are equal. Then this space of functions defiend by polynomials of degree at most 2, actually has dimension 2.

I.e. as functions 1,X,X^2 are not independent. so i omitted this part of the discussion as subtler than the easy spanning question.