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

Prove part (3) of Theorem 4.2.7.

$k \mathbf{0}=\mathbf{0}$

Calculus 3

Chapter 4

Vector Spaces

Section 2

Definition of a Vector Space

Vectors

University of Michigan - Ann Arbor

University of Nottingham

Idaho State University

Boston College

Lectures

02:56

In mathematics, a vector (…

06:36

05:14

Prove part (6) of Theorem …

02:33

Prove Theorem $2.3,$ part …

02:03

04:42

Prove Theorem 3

15:11

04:06

(a) Prove Theorem 4, part …

01:45

Prove Theorem 7.

00:45

Prove Theorem 5.3

00:58

Prove Theorem 2.3

04:27

Provea) part (ii) of T…

Okay, so we're proving part three of the're, um 4.2 point seven. Okay. Park three of theorem 4.2 point seven says okay, times the zero of actor E equals the zero vector for all scale. Er's que are an element of f Let vb the vector space over f All right, So, um, we're going to let V being arbitrary element in vector space. I'm trying to follow the, um, proof of to because it says using the fact that zero equals zero plus zero. So maybe we will go back a little bit here. We know that the zero factor equals zero vector plus the zero vac ter because of Axiom five, which says that if you start with a vector and you add this euro vector to it, you end up with the original vector, okay? And we then can re can multiply both sides of the equation times a constant so okay, times the zero vector equals k times the zero factor, plus the zero vector. And now according Teoh Axiom, nine k times the zero vector plus the zero factor is que times zero vector plus k times. The zero factor all right. So now, according Teoh Axiom six, there are additive. Inverse is so I can take Okay, Times the zero vector and odd. The opposite of K times the zero vector on both sides of the equation. Que times zero vector plus k times the zero factor plus negative k times the zero factor. And now, according to the associative property, um, of addition, that's a four. I can rewrite this as k zero vector plus que zero vector plus negative k zero vector. But, um additive inverse K zero vector plus negative k zero of actor, um would be zero. Interesting, but I'm not sure that's what I want to dio. So let's think about. Okay, So what I'm going to do is I'm going Teoh, um, factor out the K that's going to be, Ah, Axiom nine. And I'm going to do it on both sides. So that's going to give me que zero vector minus zero vector que zero vector. Okay, zero vector minus zero factor. So now according Teoh. All right, I'm gonna back up a little. I think I made little bit of ah, directional air there. We see that on the left side. Those are, um, additive Inverse is so according to a six. Um Okay. Yeah. Ah, que times zero vector plus negative. Kate M zero vector is just a zero vector. Um, same on this side. We see the same thing. So that's the zero factor. And so now, um, a five. So that was a six. Now we're doing a five on the right side, que times zero factor plus zero vector is just k times the zero vector. And so the zero vector equals k times the zero vector. And so that's what we've proven.

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…

Prove part (6) of Theorem 4.2.7.

Prove Theorem $2.3,$ part (ii).

Prove Theorem $2.3,$ part $(v)$.

(a) Prove Theorem 4, part 3.(b) Prove Theorem 4, part 5.

Provea) part (ii) of Theorem 4 .b) part (iii) of Theorem 4.

05:16

Determine the null space of $A$ and verify the Rank-Nullity Theorem.$$A=…

08:16

Let $A$ be an $m \times n$ matrix with colspace $(A)=$ nullspace(A). Prove t…

02:51

Determine all values of $a$ for which$$\left[\begin{array}{lllll}1 &…

04:50

Use the result from the previous problem to solve the given differential equ…

09:32

Let $A$ be an $n \times n$ matrix with rowspace( $A$ ) $=$ nullspace( $A$ ).…

12:54

Let $V$ be an inner product space with vectors $\mathbf{v}$ and $\mathbf{w}$…

02:19

Determine the inner product of the given vectors using (a) the inner product…

04:54

Determine the component vector of the given vector in the vector space $V$ r…

00:32

determine whether the given set of vectors is linearly independent in $P_{2}…

01:50

Decide (with justification) whether $S$ is a subspace of $V$$$V=\mathbb{…

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.