Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

Mark each statement True or False. Justify each answer.a. If $\mathbf{f}$ is a function in the vector space $V$ of all real-valued functions on $\mathbb{R}$ and if $\mathbf{f}(t)=0$ for some $t,$ then $\mathbf{f}$ is the zero vector in $V$b. A vector is an arrow in three-dimensional space.c. A subset $H$ of a vector space $V$ is a subspace of $V$ if the zero vector is in $H .$d. A subspace is also a vector space.e. Analog signals are used in the major control systems for the space shuttle, mentioned in the introduction to the chapter.

a. false.b. true.c. false.d. true.s. false.

Calculus 3

Chapter 4

Vector Spaces

Section 1

Vector Spaces and Subspaces

Vectors

Johns Hopkins University

Campbell University

Oregon State University

Baylor University

Lectures

02:56

In mathematics, a vector (…

06:36

04:11

Mark each statement True o…

06:10

In Exercises 19 and 20, V …

09:01

In Exercises 21 and $22,$ …

03:12

$\mathcal{B}$ and $\mathca…

05:09

In Exercises 19 and $20,$ …

02:09

In Exercises 15 and $16,$ …

01:35

In Exercises 17 and $18,$ …

14:16

03:54

02:07

In Exercises 21 and 22, ma…

Hello. For this question, we just need to answer five basic, true and false questions. Vector spaces. Well for basic, true and false questions about vector spaces on, then, a question about analog control systems on the space shuttle Andi Zehr Just thio test to make sure that, yeah, your your knowledge of definitions and kind of the weird border cases is where it needs to be. So for the first question, it's it. It's is a function F 50 that is equal to zero for some tea. Is this the zero vector? Where are vector space? Is the set of all real valued functions on the real line and the answer to this is false. Hopefully, this should be pretty intuitive. The zero vector when your vector space is the set of all real value function should be zero function. The names kind of work out. But the reason why is if you eyes actually in the definition of what this zero vector is, zero vector is a vector, such that for any other vector in the vector space. This relationship holds, um, this pretty kind of elementary definition of what a narrative zero is, and a good counter example to this is the sine function. So I think of sine of t and I add it and I add say, uh what za good function say one one is a real valued function and I add it Thio I had sign of tea which is zero at zero So it meets this condition over here. This does not equal one again like ah supposed zero vector should. So this shows that this is in fact false. You need thio be equal to zero everywhere along the rial line. So the second question is is a vector in three dimensional space just a narrow on that is true So this is kind of geometric interpretation. So in three dimensional space we write factors as three arbitrary constants A B and C O r you know, not not arbitrary constants but three particular constants a, B and C on This translates to some vector in three space. If I could get my drawing right, um, where the first coordinate is through the length of the the amount of the vector in the X direction. Second coordinate is the amount of vector in the Y direction and then the third coordinates the amount of actor in the Z direction. Um, so, yes, this is the correct physical interpretation of vectors in three space. So the third true and false question asks about is, uh if I have a subset of a vector space and I know that the zero vector is within this space Does this imply that h is a vector space? Um, and the answer is no. There are three conditions that a subset of a vector space has to meet to be considered a subspace. Um, the first is this condition that it contains the zero vector. The 2nd and 3rd are that they are closed under arbitrary edition of elements in this space and, um, multiplication by arbitrary constants. Eso ah, good counter example to this is if you think about our to and which is clearly a vector space, then if I think of the the unit disk which is the set of all elements, um such that this relationship holds X squared plus y squared less than or equal to one. This contains the zero vector or which is just the point which of the origin written as a vector. But it is not a vector space. Because if I if I take a vector in this space and I multiplied by by a large enough constant, it will escape this space. Eso It's not closed under multiplication by arbitrary constants. It's also not closed under addition Ah, vectors in space. Like if I take the vectors 10 and add it to the vector 01 this new vector V does not satisfy this relation show. It's not in this space. H so h is not necessarily of a subspace. All subspace is of a vector space satisfied this property. But not everything that satisfies this property is a subspace of vector space. So next question is, um, kind of related to that, which is is the subspace of a vector space. So if I have a similar relationship, I know that h is a subspace, and I know that V is a vector space. Does this imply that H is a vector space? And the answer is yes or true on. And this is, um, kind of testing back when you were developing three idea of subspace is is that say, Hey, if I If I have this this thing that sitting inside of, ah, set that I know a lot of information about. Then I should not have to go through and check everything again because there's like a dozen or so conditions that something has to meet to be a vector space. There's three that it has to meet to be a subspace, but because all of the the other properties of vector space are carried over by this relationship, just the fact that it's a simple subset. The only thing stopping, ah subset from being a vector space in and of itself are the three conditions the it has zero closed under arbitrary edition of actors as close under multiplication by arbitrary constants of actors. But yes, if something is a subspace, it is considered a vector space by itself, in fact, that the definition of a subspace is just, ah, vector space contained within another, another vector space. The final question is book dependent, not math dependent, which is, um, our analog systems used in major control systems for the subs for the space shuttle, it's mentioned in the introduction of the chapter. If you go back and read it, you'll find out that the answer is no. Analog is not used in the space shuttle anymore. That is false, all right, and that's it.

View More Answers From This Book

Find Another Textbook

In mathematics, a vector (from the Latin word "vehere" meaning &qu…

In mathematics, a vector (from the Latin "mover") is a geometric o…

Mark each statement True or False. Justify each answer.a. A vector is an…

In Exercises 19 and 20, V is a vector space. Mark each statement True or Fal…

In Exercises 21 and $22,$ mark each statement True or False. Justify each an…

$\mathcal{B}$ and $\mathcal{C}$ are bases for a vector space $V$ Mark each s…

In Exercises 19 and $20,$ all vectors are in $\mathbb{R}^{n} .$ Mark each st…

In Exercises 15 and $16,$ mark each statement True or False. Justify each an…

In Exercises 17 and $18,$ mark each statement True or False. Justify each an…

In Exercises 17 and $18,$ all vectors and subspaces are in $\mathbb{R}^{n} .…

Mark each statement True or False. Justify each answer on the basis of a car…

In Exercises 21 and 22, mark each statement True or False. Justify each answ…

00:29

Compute the determinants of the elementary matrices given in Exercises $25-3…

02:13

In Exercises $15-18,$ find a basis for the space spanned by the given vector…

01:15

Exercises $23-26$ concern a vector space $V,$ a basis $\mathcal{B}=$ $\left\…

03:28

Combine the methods of row reduction and cofactor expansion to compute the d…

03:48

In Exercises 13 and 14, find a basis for the subspace spanned by the given v…

03:11

Find the determinants in Exercises 5–10 by row reduction to echelon form.

02:55

For the matrices in Exercises $17-20,(a)$ find $k$ such that Nul $A$ is a su…

01:16

Let $W$ be the set of all vectors of the form shown, where $a, b,$ and $c$ r…

01:09

Compute the determinants in Exercises $9-14$ by cofactor expansions. At each…

01:12

In Exercises $19-24,$ explore the effect of an elementary row operation on t…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.