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

Watch this step-by-step video, matched to your homework problem.

Try Numerade Free for 30 Days

Like

Report

In Exercises 29 and $30, V$ is a nonzero finite-dimensional vector space, and the vectors listed belong to $V$ . Mark each statement True or False. Justify each answer. (These questions are more difficult than those in Exercises 19 and $20 .$ )a. If there exists a linearly dependent set $\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{p}\right\}$ in $V$then $\operatorname{dim} V \leq p$b. If every set of $p$ elements in $V$ fails to span $V,$ then $\operatorname{dim} V>p$ .c. If $p \geq 2$ and $\operatorname{dim} V=p$ , then every set of $p-1$ nonzero vectors is linearly independent.

a) Falseb) Truec) False

Calculus 3

Chapter 4

Vector Spaces

Section 5

The Dimension of a Vector Space

Vectors

Missouri State University

Campbell University

Baylor University

Idaho State University

Lectures

02:56

In mathematics, a vector (…

06:36

04:54

In Exercises 29 and $30, V…

06:10

In Exercises 19 and 20, V …

05:09

In Exercises 19 and $20,$ …

02:07

In Exercises 21 and 22, ma…

01:35

In Exercises 17 and $18,$ …

09:01

In Exercises 21 and $22,$ …

14:16

03:54

Mark each statement True o…

06:43

In Exercises 23 and $24,$ …

03:12

$\mathcal{B}$ and $\mathca…

All right. So in this in this problem, we're gonna identify where the following statements are true, Boss. So the first statement, we have a bunch of junior dependent specters. Uh, we need to identify whether dimension off B smaller or equal to P. So actually, this statement is false. To provide a counter example, let's say we have 100 and 200 so in this case, he will be too. But these two, these two vectors. All right, so in this in this problem, we're gonna identify where the following statements are true, boss. So the first statement, we have a bunch of linear dependent specters. Uh, we need to identify whether dimension a, B R they all have dimension three. And they are actually from our three. So say are three will be be in this case. So the dimension not be iss three, it is bigger than p, which is to so this statement is false. The second statement, small early, equal to p. So actually this demon is false. To provide a counter example. Let's say we have 100 and 200 So in this case, he will be too But these two These two vectors are, um, says if every every set off P elements in the fails to span V, then dimension V is bigger than P. Well, um, that's first consider recalled dimensional to be. It is defined as the element in the set off pieces is number off sectors and, uh, basis and or so are They all have to mention three and they are actually from our three. So say are three will be in this case. So the dimension of B iss three it is bigger than he which is to. So this statement is false. The second statement, um says if every every set off p element as we recall from our textbook, this is what this will be also equal to the people columns, number of people, columns Seeing the fails to span V, then Dimension B s bigger than p. Well, um, that's first consider recalled. Dimension not to be. It is defined as the number of element in the set off basis is number, uh, vectors and, uh, basis. And also, as we recall from So what does that mean? So a basis for the column space off upper matrix is equivalent to the number of people Columns in the Matrix. This is what we when we when we read in the textbook. So that means if every set off P elements in B feels to spend be so that implies, by our assumption, our assumption. Then there will be a P boat in every column. Every column as a people are textbook. This is what this will be also equal to the people Columns, number of people, columns. So what does that mean? So a basis for the column Space off for me, shakes is equivalent to the number of people columns in the Matrix. This is what we when we when we read in the textbook. So that means if every set off P elements in beef okay, so that implies. As a result, the dimension will be equal to the number of factors rather than greater than the number of factors. So that means dimension off the is equal to P rather than bigger, bigger than p. So Stillman is false. Okay, feels to spen be so that implies, by our assumption, our assumption. Then there will be a P boat in every column, every column as a people. The third statement we have if P is bigger away for 22 and Dimension V is go to pee than every set off P minus 10 back turns his leaner independent. So again, this stain these falls to see that we can check a Kendra example say s a P equals three we consider are okay, so that implies. As a result, dimension will be equal to the number of factors other than greater than the number of factors. So that means dimension I'll be is equal to P rather than bigger, bigger than P. So Stillman is false. Okay, three. So let's find a ah sad off P minus B minus one. That is, too. Just find a set off two vectors, that is Eun, you're independent. That is meaner dependent because our statement says it. It is Leonard and independent. So we find a counter example you need to find a linear in your defendant set. So it's very easy to find. Let's say, um, ones. The third statement we have if P is bigger away for 22 and Dimension V is going to be then every set off p minus 10 back turns his leaner independent. So again, this stain these falls to see that we can check a Kendra example say s A P equals three we consider are three. There was zero and 200 See, these two factors are the New York dependent. But we only have like and this is Yeah, we only have two vectors. So dead at exactly sapi minus one. So too So that studies flyover assumption here. And we we did find a union linearly dependent factors. So So let's find a ah sad off P minus B minus one. That is, too. Just find a set off two vectors, that is, Eun, you're independent. That is very dependent because our statement says it is Leonard and independent. So we find a counter example you need to find a leaner in your defendant set. So it's very easy to find. Let's say, um one there is false

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…

In Exercises 29 and $30, V$ is a nonzero finite-dimensional vector space, an…

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

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

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

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

In Exercises 21 and $22,$ 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 23 and $24,$ mark each statement True or False. Justify each an…

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

01:52

If $A$ is a $6 \times 4$ matrix, what is the smallest possible dimension of …

03:21

Let $\left\{y_{k}\right\}$ be the sequence produced by sampling the continuo…

02:43

Given subspaces $H$ and $K$ of a vector space $V,$ the sum of $H$ and $K,$ w…

03:18

Find the general solution of difference equation $(15) .$ Justify your answe…

04:19

Let $\mathcal{B}=\left\{\mathbf{b}_{1}, \mathbf{b}_{2}, \mathbf{b}_{3}\right…

01:12

Use Exercise 27 to complete the proof of Theorem 1 for the case when $A$ is …

03:08

Define $\quad$ a linear transformation $\quad T : \mathbb{P}_{2} \rightarrow…

02:01

For each subspace in Exercises 1–8, (a) find a basis, and (b) state the dime…

00:57

Is $\lambda=-2$ an eigenvalue of $\left[\begin{array}{rr}{7} & {3} \\ {3…

06:02

In Exercises 11–16, compute the adjugate of the given matrix, and then use 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.