linearly independent subsets

Consider the following subset of P3(R) (real polynomial functions of degree
at most 3). Z := {f1; f2; f3; f4; f5} with:
$f1(x) = 1+2x-x^2+3x^3, [br]f2(x)= 2-x+x^2+x^3,[br]f3(x)= 5x-3x^2+5x^3,[br]f4(x)= 1-3x+x^2-x^3,[br]f5(x)= 4+3x-2x^2+8x^3.$
Prune Z to produce a linearly independent subset Y with Span(Z) = Span(Y). What is the dimension of Span(Z)? Is p3 an
element of Span(Z)? (Recall that p3(x) = x^3.) Extend Y to give a basis for
P3(R).

I've put the functions into a matrix and row reduced it but I do not know what to do from here .
Any help at all? Am I even supposed to row reduce them?

You need to understand what the question is asking and what the technique actually does rather than just jumping straight into the technique blindly and then the answer will be there.

Linear dependencies between elements in $Z$ are non-trivial solutions to the equation $\sum_{i=1}^5\lambda_i f_i = 0$. You can find these by writing out the equation in matrix form and solving by row reduction. Once you find a non-trivial solution - you rearrange the equation to express one vector as a combination of the others and thus may remove that vector. You then repeat until you have vectors left that are independent. In practice, once you know what you are doing you can basically do this in one step.

To complete a set of linearly independent vectors to a basis, you can do the same thing in reverse. You know the dimension of your space so you know how many vectors you need to add. Take the required number of vectors with arbitrary coefficients and then solve the associated system which will tell you how to choose the coefficients so that you get only the trivial solution i.e. so that the vectors are independent.

Hi I really don't understand. So I am supposed to row reduce the matrix, but what do I do then? I've read all of what you said but i can't make sense of it.

You need to forget about your answer for the moment.

There is no point in saying "I don't understand anything I have written down but want to continue with my solution."

Trying to do that is the reason you are stuck.

You need to go back to the beginning. This is really straightforward bookwork. The only reason you can be struggling is because you don't know about the basics of linear dependence, matrices and the link between systems of equations etc. and need to revise them.
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So, in what sense did you "put the functions into a matrix" and why did you then row reduce it?

I am not being facetious here by the way. The reason you are stuck is because you have done the above mentioned things for no reason (or presumably because you didn't read your notes thoroughly but skipped to the end to an example and tried to mimic it). If you understood what you were doing, you would then know why you did it and would therefore understand how to proceed.

So I can't help you until you either:

- tell me why you did what you already did; or
- you start from the beginning and explain clearly what you don't understand.

I have no notes and can't find any online.

That's why I am stuck lol.

I put the equations in horizontally -although I'm starting to think I should have done them vertically- and row reduced them as that was the only mathematical process I could think of other then simultaneous equations that would literally be 'pruning' Z.

I know that a matrix is linearly independent if it's determinant is non-zero

There are other things later in the question I don't understand, but I need to get my head round the early stages first.

That said, let's start from the beginning.

So you have five vectors which live in the vector space $P_3(\mathbb{R})$.

You want to remove some of those vectors to produce a set that is linearly independent but has the same span.

So, what does it mean for vectors to be linearly independent (and don't say anything about matrices unless you are able to tie it in to the question at hand i.e. the linear independence of the vectors in question)?

That is the first step. I know you said you don't have notes/didn't go to lectures and couldn't find it online but I don't believe the latter point and can't be arsed writing out definitions that you are two mouse clicks away from...

Answer that, come back and we will do the next step.

I could find it online if I knew what it were called but by googling 'pruning a subset' or different variations I couldn't define anything.

A vector(x) is linearly independent from the vectors (y and z) if x can not be written as a linear combination of y and z.

Pruning isn't being used in any technical sense here. It means, as I said in the previous post, removing some of the vectors from that set so that the set is linearly independent.

Once again you are jumping the gun and forgoing the fact that you haven't understood the preceding part of the question.



Ok. And what does it mean for a set of any number of vectors to be linearly independent?

A set of vectors is linearly independent if none of the vectors are scalar multiples of each other.

Ok, so a set of vectors is linearly independent if each pair of vectors in it are linearly independent i.e. they aren't scalar multiples of each other.

Another way to say this (and useful from the point of view of this question) is that a set of vectors $v_1,\ldots,v_n$ is linearly independent if the only linear combination $\lambda_1v_1 + \cdots + \lambda_nv_n$ of them equal to zero is the trivial one i.e. the one where each of the $\lambda_i$ is zero.

If a set of vectors is not linearly dependent, it is called linearly dependent. A linear dependence between a set of vectors just means writing one vector as a linear combination of the others i.e. it is a nontrivial solution to the equation:

$\lambda_1v_1 + \cdots + \lambda_nv_n=0$.

So, you want to find linear dependencies and remove all vectors that are linearly dependent on the remaining ones until you are left only with a linearly independent set whose span as the same as the original.

So what is the span of a set of vectors?

How would you go about finding solutions to the above equation? (and be precise here - don't just say "write a matrix" - be specific as to how it is equivalent to the above equation)

You are still trying to jump the gun, sure you are starting to understand what any of this has to do with matrices but if you persevere a tiny bit more, you can say precisely which matrix we should look at and why. Remember that jumping straight to the matrix without having anything other than a vague memory of seeing them in this scenario before was why you were stuck.

I really do not know how I would go about solving that equation. I guess that's my problem here.

Try writing your vectors in terms of some basis. Do you know what a basis is? Do you know what the standard basis is for the space under consideration (i.e. $P_3(\mathbb{R})$?

How would you use matrices to write, for example:

$\lambda_1\cdot \begin{pmatrix}1 \\ 2\end{pmatrix} +\lambda_2\cdot \begin{pmatrix}3 \\ 4\end{pmatrix} = \begin{pmatrix}0 \\ 0\end{pmatrix}$


(to be honest - I am staggered that you have done a linear algebra course and have seen matrices without being told how they can be used to express systems of linear equations)

The difficulty is I just can't understand all of this. I can do a method if I see an example but I can't read all of this and understand it. I have to see someone do it and then I can do it.

This isn't helping me at all so I appreciate the effort but I'm just getting more confused. I'll keep looking for examples. Thanks for the help.

LOL. You were actually making progress here because you started to learn something. You need to have the patience to not understand everything in the time it takes to read it. If you are working through this and found something you couldn't understand, then that is great because you are finally in a position to ask a decent question which you can be helped with

When you were looking at examples and trying to copy them without thinking what anything actually meant you got miserably stuck.

So good luck with that.

I mean what is it you don't understand about "do you know what a basis is?" you either know it or you don't... sure you may or may not understand the concept or the relevance yet but just take it one step at a time. Your problem is unwillingness to work through slowly and patiently. It is ironic because you are willing to waste time looking for shortcuts and trying to answer the question without understanding what it is asking but in reality, you would get there quicker if you just accepted that that is the only way you will be able to do this and buckled down and looked up the words that you don't know.

Do you fail to understand that you will not be able to do any manipulations with matrices to solve problems until you can do the much easier fundamentals?

If you don't know how to write a linear system in two unknowns as a matrix equation then you aren't ready for this question.

Look I've been talking to you for two days and if I wrote down everything you taught me in the same question in an exam I would get 0 marks because I would have answered none of the question. I was trying to be polite, but you have worn down my patience with your condescending tone. You have wasted enough of my time. Stop talking down to me, if you want to help someone that's not the way to do it.

You have been at it for two days because you are waiting for a miracle and don't have the patience to look up the words you don't know the meaning of.

You are beyond help.

I've looked up the definitions for all the words I didn't know the meaning too. Unfortunately the question isn't defining a bunch of words it's doing a process and towards that all you have been is a liability.

Look I've been talking to you for two days and if I wrote down everything you taught me in the same question in an exam I would get 0 marks because I would have answered none of the question.