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Determine if the given set is a subspace of $\mathbb{P}_{n}$ for an appropriate value of $n .$ Justify your answers.All polynomials of the form $\mathbf{p}(t)=a+t^{2},$ where $a$ is in $\mathbb{R}$

the given set is not closed under vector addition.

Calculus 3

Chapter 4

Vector Spaces

Section 1

Vector Spaces and Subspaces

Vectors

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Harvey Mudd College

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Lectures

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In mathematics, a vector (…

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Determine if the given set…

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Express $S$ in set notatio…

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What is the dimension of t…

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Decide (with justification…

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in this video, we're gonna be solving problem number six of section 4.1. And it asks us to determine if the given set is a subspace of piece of men for inappropriate for any appropriate value of n. Um, And the given polynomial we have is p f t equals a plus t squared where a is any real number. So this is pretty simple. Um, for example, it could be any real numbers. So let's just choose four plus t squared as our polynomial for now. Um, And for for this particular set of calling almost to be in peace of end, you have to be closed under scaler multiplication. So the determine if it is closed under Scott wealth creation, we can just multiply by like like any number. Let's say three. So we got 12 plus three t square. Um, for for this example polynomial, um, we see that 12 is still in still a part of all real numbers, because is an all real numbers. But three t squared is doesn't fit fit our original definition of pft because gift, he is a plus t squared where he has a coefficient of one. But here. He has a quick, efficient of three. So this set is not close under skill multiplication. So it's not a subspace of piece of in yeah, and that's that's basically it.

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