explain-why-the-space-mathbbp-of-all-polynomials-is-an-infinite-dimensional-space

Name: explain why the space mathbbp of all polynomials is an infinite dimensional space
Uploaded: 2021-01-13
Duration: 5 min 48 s
Channel: Numerade
Description: Explain why the space $\mathbb{P}$ of all polynomials is an infinite-dimensional space.

Question

Explain why the space $\mathbb{P}$ of all polynomials is an infinite-dimensional space.

Anthony Ramos · Accepted Answer

Hello there. Okay, So for this exercise, we need to prove that the set off polynomial. Sorry. Hey, this is our I&#x27;m meaning. This set the set off polynomial actually corresponds. Finito, dimensional vector space. So how to prove that? Okay, so it could be quite a interesting, but how is defined the set of polynomial, So basically, we can expand. Okay, So instead of polynomial is basically the span off one x x two extra square here X to the square X to the third you want until X m. Okay. So we can define, um, integral end. So in this case, and corresponds to an element off natural. Ah, on by the the the space off phenomenally defined by the span of these. So just to recall that the spun means Elinor combinations off these elements here on, it&#x27;s clear that that corresponds to a paranormal off degree and right, So, for example, let&#x27;s say that n equals 22 We got a X square plus b x plus c. So here we got a linear combination off this elements here until degree to because we fix it and equal to two okay on. In this case, we&#x27;re responding. The set off point, however, were interested on spend the whole set off polynomial and actually we confined Objection. So let&#x27;s define the following the following function. So let&#x27;s say we can define a function from the natural to the set off polynomial B n. That is really easy surgeries people to the P. That actually is really easy to construct this because we&#x27;re going to take for any end on the natural on. We&#x27;re going to translate into X to the end. Okay. And we basically were constructing polynomial we find, ah, function that take any any value off here from the natural here. So let&#x27;s say 123 on we&#x27;re map into the space off polynomial we&#x27;re taking one. For example. On we get X to the one is an element off. The point on this again for two we can map two x square for three is the same X three okay on this F form by ejection because for every number for every element here there exists one element on the side of Paul Anonymous. So is one by one, correspondence on for ah, on every every every element. Here is map to the side of polygamy is very subjective. So this function f is onto us objective. So that means that f responsive rejection So great. So we have defined that f is a rejection on Actually, if you notice f form a basis for the set off for a Nahmias Why? Because we have a state before we have already see that this span at the set, the space of pollen animals. So basically, we can say a basis for P will be the basis for the space. P will be the set off functions off end. Such that end is an element off naturals. What I am putting into these ways basis is that fn Well, what is f one If one corresponds to x f zero corresponds toe one and so on. So basically here in this said I&#x27;m putting one x x square to the third until extend But here we got something And is any element off the natural aunt, we know that the set off natural is an infinite set is a new infinite comfortable on we have fine Ah, by ejection f from n to the set of polynomial. So that means on also f is rejected, so f is by objective. So that means that and at least, um, this off the set off Paula normals is, uh, has Dimension has the same dimension that end and in this case and actually objectives of have so have the same dimension on and is an infinite set. Soapy should be on infinite dimensional space as well, and that completes the proof.

Amy Jiang · Answer

Okay, So we&#x27;re given our fallen function and are also given that the limit as X approaches a from the left hand side to from the left hand side, that is X less any threats G of X. That&#x27;s equal to Gov. And we know that both geo backs and each of X or polynomial okay and were asked to find the limits as X approaches A from the rights. So that&#x27;s a value greater than a So that&#x27;s this h of X. And since, um, each of a X is a polynomial, the limits is equal to H. Evaluated ads are points, which is a

Katelyn Chen · Answer

for us explain why all functions must have raised 1920 to infinity. Well, if you think about it, all functions only have two scenarios. So a straw to simple graphs. And so either as X approaches negative infinity, half of axel approach infinity. That&#x27;s X approaches infinity of fact. So push night affinity or else as X approaches that Vinny Effectual personal property and its X approaches. If in the effort back So a project Night of brandy. And you know this that both of these graphs have arranged. I&#x27;ve been fending Tiffany this because with a law degree, you will always have one direction pointing up. No one on either side will be pointing the opposite direction. And so no matter what, it will forever reach Negative infinity! You&#x27;re fiber. Reach the range. Negative. Infinity to infinity

Katelyn Chen · Answer

explain what oh polynomial is was in awe. Degree must have at least 10 So that means they must cross the X axis as least one time. So you know that any odd function and every single our function has the range native defending to infinity. And because the ranges native infinity to infinity, then you know that it must so it must put must and difficulty must cross the ex access, which creates zero.

Heather Zimmers · Answer

in this problem, we&#x27;re figuring out why does a function like F of X equals four have degree zero. Remember that the degree is the greatest exponents on the variable. If you see f of X equals four, you can think of it as f of X equals four times one. And you can think of that as f of X equals four times X to the zero because X to the zero is one. So if you think of it that way, you can see that you&#x27;re exponents on X is zero. And that&#x27;s the only one in the whole function. So that must be the greatest. That&#x27;s why it has degree zero. Similarly, you could rewrite f of X equals X plus five, and you could write it as f of X equals X to the first plus five x to the zero. And now you&#x27;re looking for the greatest exponents on your variable. And so the greatest exponents is the one. It&#x27;s greater than zero. So it has degree one

Ankit Gupta · Answer

so hello, Strain on Already left. In this question, the question is extremely. Why a partner Johnson off even degree with domain. My last unified toe in front cannot be 11 December. The even big difference in Minot, he went big difference for acquired it. If you see the even Vincent Graff, you see the graph. The graph, either a port opening our downward opening are If fourth degree, then graph. Maybe like this. Our growth may go like this for even Pinson. Always this endpoint like same side, a pork, our dawn work so not object in this situation. And you&#x27;ll find by a hard in the line test dental line test holding the line a hard a gentle line in dissect more than more than one point. Therefore, it is not 1 to 1 person and for the V person with this thing, explain a white Some cases if boiler or degree or degree graph hard to get it off a straight line. Go like this. This is X axis. Go like this are hard degree graph. Go like this in discusses the horizontal cut only one. So this is 11 This is also one but In this case, the graph is in order. One winning original line got many times three times. This is also grabbed for it. Order falls and art upon since this book tell on head are and one light in different side off the X axis. So after in some cases like this is like this and this harden the like cut more than one point. So, in some cases, a holy, gentle line in the second, uh, more than more than one point. Therefore, it is not necessary. It is not necessary that all or big refunds, one school or

Problem

Show that the space $C(\mathbb{R})$ of all contin…

Problem 27 Easy Difficulty

Explain why the space $\mathbb{P}$ of all polynomials is an infinite-dimensional space.

Answer

Related Courses

Linear Algebra and Its Applications

Related Topics

Discussion

Top Calculus 3 Educators

Heather Z.

Kayleah T.

Samuel H.

Michael J.

Calculus 3 Courses

Vectors Intro

Vector Basics - Example 1

Recommended Videos

Watch More Solved Questions in Chapter 4

Video Transcript

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Heather Z.

Kayleah T.

Samuel H.

Michael J.

Vectors Intro

Vector Basics - Example 1

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

Linear Algebra and Its Applications

Heather Z.

Kayleah T.

Samuel H.

Michael J.

Vectors Intro

Vector Basics - Example 1

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Heather Z.

Kayleah T.

Samuel H.

Michael J.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications