[LinAlg] Set Notation, Subspaces, Bases, Ranges, & Kernels

[bornofflame]

Homework Statement

I don't want to clog up the forums with a few "small" problems so I am lumping them together here.

2. Let $T:P^1 → R \text { be given by } T(p(x)) = \int^b_a p(x)dx$. Describe Ker(T) using set notation.

3. Let $H = \left\{f ∈ C[a, b] | f'(x) ≥ 0 ~\text for ~a<x<b\right\}$. Determine if H is a subspace of C[a, b], and if so, prove it. If H is not a subspace explain why not.

4. Let $H = \left\{f ∈ C[-1, 1] | f(0) = 0\right\}$. Determine if H is a subspace of $C[-1, 1]$, and if so, prove it. If H is not a subspace explain why not.

5. Let $H = Span \left\{cos x, sin x\right\}. \rm Let ~B = \left\{cos x, sin x\right\}$ be the standard basis for H. Also let D = {3cos x - sin x, 2cos x + sin x} be another basis for H. Let f(x) = 5cos x + 10sin x be an element of H.
a. Determine $\left[ f \right]_B$
b. Determine the matrix P such that $P\left[ f \right]_B = \left[ f \right]_D$.
c. Determine $\left[ f \right]_D$

6. Let $T:P^2 \rightarrow P^2$ be defined by $T(p(x)) = \frac d {dx} p(x)$
i. Describe the range of T.
ii. Describe the kernel of T.

7. Let $T:P^1 \rightarrow P^1$ be defined by $T(p(x)) = \int^1_0 p(x){dx}$
i. Describe the range of T.
ii. Describe the kernel of T.

Homework Equations

The Attempt at a Solution

2. The kernel is the set of solution in $P^1$ that map to the 0 vector in ℝ, so:
$Ker(T) = \left\{x | T(x) = 0\right\}$

3. To be a subspace H must contain the 0 vector and be closed under vector addition and scalar multiplication:
ii. Closed under vector addition: (f + g)(x) = [F(b) - F(a)] + [G(b) - G(a)].
I'm not really sure where to go after this, however,

4. To be a subspace H must contain the 0 vector and be closed under vector addition and scalar multiplication:

i. f(0) = 0, Since H only contains the functions that are equal to 0 when their input is 0, it definitely contains the 0 vector.

ii. Closed under vector addition: (f + g)(0) = 0 + 0 = f(0) + g(0).
Not really sure that this holds water, to be honest. To show that it's closed, I need more general evidence.

iii. Closed under scalar multiplication: f(c0) = 0 = cf(0)
∴ H is a subspace.

5.
a. $\left[ f \right]_B = \left[ c_1 ~ c_2 \right] \rightarrow c_1 = 5, c_2 = 10 \rightarrow \left[ f \right]_B = \begin{bmatrix} 5 \\ 10 \end{bmatrix}$

b. $P = ? \text { such that } P \left[ f \right]_B = \left[ f \right]_D$
$P \begin{bmatrix}5 \\ 10 \end{bmatrix} = \left[ f \right]_D$
$P = \begin{bmatrix}3 & 2 \\ -1 & 1 \end{bmatrix}$

c. $\left[ f \right]_D = ?$
Steps:
1. Construct $P_{B \leftarrow D}$:
Done in part b: $P = \begin{bmatrix}3 & 2 \\ -1 & 1 \end{bmatrix}$

2. Find $P^1_{B \leftarrow D}$:
Per the calculator: $P^1_{B \leftarrow D} = \begin{bmatrix} 0.2 & -0.4 \\ 0.2 & 0.6 \end{bmatrix}$

3. Solve for $\left[ x \right]_D$:
$\left[ x \right]_D = P^1_{B \leftarrow D} \left[ x \right]_D = \begin{bmatrix} 0.2 & -0.4 \\ 0.2 & 0.6 \end{bmatrix} \begin{bmatrix}5 \\ 10 \end{bmatrix} = \begin{bmatrix}-3 \\ 7 \end{bmatrix}$

6.
a. Ker(T) is the set of all p(x) in $P^2$ that are mapped onto p(x) = 0
or
$Nul T = \left\{p(x) | p(x) \in P^2 and T(p(x)) = 0 \right\}$

b. Range T is the set of all T(p(x)) in $P^2$ that are solutions for T(p(x)) = b

7.
a. Range T is the set of numbers resulting from the mapping from $P^1 \rightarrow P^1$
b. Ker T is the set of all $T(p(x))$ such that $\int^1_0 p(x){dx} = 0$

 

[PeroK]

Homework Statement

I don't want to clog up the forums with a few "small" problems so I am lumping them together here.

2. Let $T:P^1 → R be given by T(p(x)) = \int^b_a p(x)dx$. Describe Ker(T) using set notation.

3. Let $H = \left\{f ∈ C[a, b] | f'(x) ≥ 0 ~\text for ~a<x<b\right\}$. Determine if H is a subspace of C[a, b], and if so, prove it. If H is not a subspace explain why not.

Homework Equations

The Attempt at a Solution

2. The kernel is the set of solution in $P^1$ that map to the 0 vector in ℝ, so:
$Ker(T) = \left\{x | T(x) = 0\right\}$

3. To be a subspace H must contain the 0 vector and be closed under vector addition and scalar multiplication:
ii. Closed under vector addition: (f + g)(x) = [F(b) - F(a)] + [G(b) - G(a)].
I'm not really sure where to go after this, however,

You should only post one question per thread. And, these don't look like serious attempts at a solution. I suggest you concentrate on one problem at a time.

For 2), my guess is that you are supposed to describe the functions in the Kernel, not simply write down the defintion of the Kernel. Moreover, you've given a subset of $\mathbb{R}$ as the Kernel, where it should be a subset of the polynomial in $P^1$.

For 3) you seem to be integrating for some reason? You need to think much more carefully about what this question is asking.

 

[bornofflame]

You should only post one question per thread. And, these don't look like serious attempts at a solution. I suggest you concentrate on one problem at a time.

Ah! Apologies.
I'll remember to do that.

For 2), my guess is that you are supposed to describe the functions in the Kernel, not simply write down the defintion of the Kernel. Moreover, you've given a subset of $\mathbb{R}$ as the Kernel, where it should be a subset of the polynomial in $P^1$.

Hmm. Can you clarify on what you mean by describe the functions? I think I have an idea as follows, but I'm not sure.
$p(t) \in \mathbb{P}^1 \text { where } p(t) = a_0 + a_1t = 0$ where t are the weights
$t^0 \begin{bmatrix}a_0 \end{bmatrix} + t^1 \begin{bmatrix} a_1 \end{bmatrix} = 0, \text { so that } A=\begin{bmatrix} a_0 & a_1~ |~ 0 \end{bmatrix}$ has only the trivial solution.

$Ker T = \left\{ p(x) : p(x) \in \mathbb{P}^1 \text { and } Ap(x) = 0 \right\}$

For 3) you seem to be integrating for some reason? You need to think much more carefully about what this question is asking.

I'll leave this and the others alone for now to focus on two then post them separately after I figure that one out.

 

[PeroK]

I think you need to identify the functions whose integral is 0.

What does that imply about $a_0, a_1$?

 

[bornofflame]

Realized that I was mixing in t's and x's. Hopefully that hasn't caused any confusion.

That aside, I realized that $\int_0^1 \frac 1 p x^q - \frac 1 r \frac d {dx} \text { where } p \in \mathbb{R} \text { and } r = p \left( q + 1 \right)$ would result in a value of 0, and that's what I could come up with my sleep addled mind. So, possibly, likely, there are others that I have not yet conceived.
I think, then, that it doesn't matter what values of x we have we are dealing with the zero vector, where $a_0 = 0, \text { and } a_1 = 0$.

 

[PeroK]

Realized that I was mixing in t's and x's. Hopefully that hasn't caused any confusion.

That aside, I realized that $\int_0^1 \frac 1 p x^q - \frac 1 r \frac d {dx} \text { where } p \in \mathbb{R} \text { and } r = p \left( q + 1 \right)$ would result in a value of 0, and that's what I could come up with my sleep addled mind. So, possibly, likely, there are others that I have not yet conceived.
I think, then, that it doesn't matter what values of x we have we are dealing with the zero vector, where $a_0 = 0, \text { and } a_1 = 0$.

Do you know how to integrate a linear polynomial?

You are trying to identify polynomials which integrate to 0. Values of x are not relevant.

It's the values of $a$ and $b$ that are important.

 

[bornofflame]

Correct me if I'm wrong but the integral of any linear polynomial would be of the form $\frac 1 2 ax^2 + c$ since a linear polynomial would be something like 5x + 2, y - 7, etc.

I wasn't trying to be specific in my example, I think that was more for my benefit because I was thinking that there were none at first and then realized that I was wrong.

 

[PeroK]

Correct me if I'm wrong but the integral of any linear polynomial would be of the form $\frac 1 2 ax^2 + c$ since a linear polynomial would be something like 5x + 2, y - 7, etc.

I wasn't trying to be specific in my example, I think that was more for my benefit because I was thinking that there were none at first and then realized that I was wrong.

A definite integral is a number, not a function of $x$. And, in fact, your integration is wrong.

You need to calculate:

$\int_a^b \alpha x + \beta dx$

Set it to 0 and solve for $\alpha, \beta$.

I used Greek letters as I prefer to use different letters for variables and constants, which are $a,b$ in this case.

 

[Ray Vickson]

Homework Statement

I don't want to clog up the forums with a few "small" problems so I am lumping them together here.

2. Let $T:P^1 → R \text { be given by } T(p(x)) = \int^b_a p(x)dx$. Describe Ker(T) using set notation.

3. Let $H = \left\{f ∈ C[a, b] | f'(x) ≥ 0 ~\text for ~a<x<b\right\}$. Determine if H is a subspace of C[a, b], and if so, prove it. If H is not a subspace explain why not.

4. Let $H = \left\{f ∈ C[-1, 1] | f(0) = 0\right\}$. Determine if H is a subspace of $C[-1, 1]$, and if so, prove it. If H is not a subspace explain why not.

5. Let $H = Span \left\{cos x, sin x\right\}. \rm Let ~B = \left\{cos x, sin x\right\}$ be the standard basis for H. Also let D = {3cos x - sin x, 2cos x + sin x} be another basis for H. Let f(x) = 5cos x + 10sin x be an element of H.
a. Determine $\left[ f \right]_B$
b. Determine the matrix P such that $P\left[ f \right]_B = \left[ f \right]_D$.
c. Determine $\left[ f \right]_D$

6. Let $T:P^2 \rightarrow P^2$ be defined by $T(p(x)) = \frac d {dx} p(x)$
i. Describe the range of T.
ii. Describe the kernel of T.

7. Let $T:P^1 \rightarrow P^1$ be defined by $T(p(x)) = \int^1_0 p(x){dx}$
i. Describe the range of T.
ii. Describe the kernel of T.

Homework Equations

The Attempt at a Solution

2. The kernel is the set of solution in $P^1$ that map to the 0 vector in ℝ, so:
$Ker(T) = \left\{x | T(x) = 0\right\}$

3. To be a subspace H must contain the 0 vector and be closed under vector addition and scalar multiplication:
ii. Closed under vector addition: (f + g)(x) = [F(b) - F(a)] + [G(b) - G(a)].
I'm not really sure where to go after this, however,

4. To be a subspace H must contain the 0 vector and be closed under vector addition and scalar multiplication:

i. f(0) = 0, Since H only contains the functions that are equal to 0 when their input is 0, it definitely contains the 0 vector.

ii. Closed under vector addition: (f + g)(0) = 0 + 0 = f(0) + g(0).
Not really sure that this holds water, to be honest. To show that it's closed, I need more general evidence.

iii. Closed under scalar multiplication: f(c0) = 0 = cf(0)
∴ H is a subspace.

5.
a. $\left[ f \right]_B = \left[ c_1 ~ c_2 \right] \rightarrow c_1 = 5, c_2 = 10 \rightarrow \left[ f \right]_B = \begin{bmatrix} 5 \\ 10 \end{bmatrix}$

b. $P = ? \text { such that } P \left[ f \right]_B = \left[ f \right]_D$
$P \begin{bmatrix}5 \\ 10 \end{bmatrix} = \left[ f \right]_D$
$P = \begin{bmatrix}3 & 2 \\ -1 & 1 \end{bmatrix}$

c. $\left[ f \right]_D = ?$
Steps:
1. Construct $P_{B \leftarrow D}$:
Done in part b: $P = \begin{bmatrix}3 & 2 \\ -1 & 1 \end{bmatrix}$

2. Find $P^1_{B \leftarrow D}$:
Per the calculator: $P^1_{B \leftarrow D} = \begin{bmatrix} 0.2 & -0.4 \\ 0.2 & 0.6 \end{bmatrix}$

3. Solve for $\left[ x \right]_D$:
$\left[ x \right]_D = P^1_{B \leftarrow D} \left[ x \right]_D = \begin{bmatrix} 0.2 & -0.4 \\ 0.2 & 0.6 \end{bmatrix} \begin{bmatrix}5 \\ 10 \end{bmatrix} = \begin{bmatrix}-3 \\ 7 \end{bmatrix}$

6.
a. Ker(T) is the set of all p(x) in $P^2$ that are mapped onto p(x) = 0
or
$Nul T = \left\{p(x) | p(x) \in P^2 and T(p(x)) = 0 \right\}$

b. Range T is the set of all T(p(x)) in $P^2$ that are solutions for T(p(x)) = b

7.
a. Range T is the set of numbers resulting from the mapping from $P^1 \rightarrow P^1$
b. Ker T is the set of all $T(p(x))$ such that $\int^1_0 p(x){dx} = 0$

One problem per posting, please.
Also: in problem (5), please use LaTeX correctly: you should write
$H = \text{Span} \left\{\cos x, \sin x\right\}$ instead of what you wrote. You do that by putting the word "Span" in a text box (that is, writing \text{Span}) and writing "\cos" or "\sin" instead of "cos" or "sin". Similarly, it is better to write $B = \{ \cos x , \sin x \}$ than to use a roman font---again by writing \cos and \sin.. LaTeX is designed to give pleasing results if you use it properly.

 

[fresh_42]

You should only post one question per thread.

...because our rules request you to do so.

And, these don't look like serious attempts at a solution. I suggest you concentrate on one problem at a time.

Good advice.

One problem per posting, please.

... which is why I close this thread now. And please, report such threads instead of answering them. Thank you.