# Need some hints on my HW about Linear functionals

• MHB
• BaconInDistress
In summary, In problem 3, what is the third derivative of a polynomial whose degree is at most 2?For problem 4, extend $v$ to a basis of the space $V$.
BaconInDistress
I not very good at using the LaTex editor, so I took a photo of my HW questions.
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For the first question, I'm not really sure how to get started, should I write out a specific case? Like what would $$\displaystyle \varphi (P)$$ be when m=1?

For the second question, I know that a linear functional have two properties, one being $$\displaystyle \varphi(u +v) = \varphi(u) + \varphi(v)$$ and the other one being $$\displaystyle \varphi(\lambda u) = \lambda\varphi(u)$$. I'm a bit confused about the mapping, is it saying that when we have a polynomial P(z), the map makes it goes to p(0) for all z?

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Perhaps you should start by defining notations, including $\mathbb{F}$, $P_m(\mathbb{F})$ and $p^{(k)}(x)$. Also, it is not very clear which of $p$'s are uppercase and which are lowercase letters. For example, in problem 1 in the left-hand side $\varphi(P)$ the letter $P$ seems to be uppercase, but in the right-hand side it is probably lowercase.

According to the textbook we use, $$\displaystyle F$$ is a field over $$\displaystyle \Bbb{R}$$ or $$\displaystyle \Bbb{C}$$. $$\displaystyle {P}_{m}(F)$$ is the polynomial space of degree m.
One of the many problem with my homework is indeed the handwriting, my professor is not very consistent with his upper and lower case letters. I just copied down what he wrote.
I think for question one, we need to prove the existence of constants $$\displaystyle {a}_{0},...,{a}_{m}$$ such that the linear functional $$\displaystyle \varphi (P) = {a}_{0}p(0) + \sum_{k=1}^{m} {p}^{(k)}(0)$$
Not really sure how to go from there

BaconInDistress said:
According to the textbook we use, $$\displaystyle F$$ is a field over $$\displaystyle \Bbb{R}$$ or $$\displaystyle \Bbb{C}$$.
"A field over $\mathbb{R}$" sounds strange. There are vector spaces and algebras over fields. Perhaps $\mathbb{F}$ is simply a field.

BaconInDistress said:
I think for question one, we need to prove the existence of constants $$\displaystyle {a}_{0},...,{a}_{m}$$ such that the linear functional $$\displaystyle \varphi (P) = {a}_{0}p(0) + \sum_{k=1}^{m} {p}^{(k)}(0)$$
Not really sure how to go from there
I think $p$ should be lowercase in the left-hand side as well.

Every functional on $\mathbb{F}^m$ has the form $\varphi((b_0,\ldots,b_m))=a_0b_0+\dots a_mb_m$ for some constants $a_0,\ldots,a_m$. Now suppose $p(t)=b_0+b_1x+\dots b_mx^m$. What are $p(0), p'(0), \ldots, p^{(m)}(0)$?

I'm a bit confused about the mapping, is it saying that when we have a polynomial P(z), the map makes it goes to p(0) for all z?
Yes, I believe the function maps a polynomial $p(z)$ to $(p(0))^2$. The question is whether it is a linear functional.

In problem 3, what is the third derivative of a polynomial whose degree is at most 2?

For problem 4, extend $v$ to a basis of the space $V$.

## 1. What is a linear functional?

A linear functional is a mathematical function that maps a vector space to its underlying field of scalars. It is a type of linear transformation that takes in a vector and outputs a scalar value.

## 2. How do I find the kernel of a linear functional?

The kernel of a linear functional is the set of all vectors in the vector space that map to the zero scalar value. To find the kernel, set the linear functional equal to zero and solve for the variables in the vector space.

## 3. What is the difference between a linear functional and a linear transformation?

A linear functional maps a vector space to its underlying field of scalars, while a linear transformation maps a vector space to another vector space. In other words, a linear functional outputs a scalar value, while a linear transformation outputs a vector.

## 4. How do I determine if a linear functional is injective or surjective?

An injective linear functional maps distinct vectors to distinct scalar values, while a surjective linear functional maps every scalar value in the underlying field to at least one vector in the vector space. To determine if a linear functional is injective or surjective, you can use the rank-nullity theorem.

## 5. Can a linear functional have more than one kernel?

No, a linear functional can only have one kernel. This is because the kernel is defined as the set of all vectors that map to the zero scalar value, and there can only be one zero scalar value in the underlying field.

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