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

Let $W$ be a subspace of $\mathbb{R}^{n},$ and let $W^{\perp}$ be the set of all vectors orthogonal to $W .$ Show that $W^{\perp}$ is a subspace of $\mathbb{R}^{n}$ using the following steps.a. Take $\mathbf{z}$ in $W^{\perp}$ , and let u represent any element of $W .$ Then $\mathbf{z} \cdot \mathbf{u}=0 .$ Take any scalar $c$ and show that $c \mathbf{z}$ is orthogonal to $\mathbf{u} .$ (Since $\mathbf{u}$ was an arbitrary element of $W,$ this will show that $c \mathbf{z}$ is in $W^{\perp} . )$b. Take $\mathbf{z}_{1}$ and $\mathbf{z}_{2}$ in $W^{\perp},$ and let $\mathbf{u}$ be any element of$W .$ Show that $\mathbf{z}_{1}+\mathbf{z}_{2}$ is orthogonal to $\mathbf{u} .$ What can you conclude about $\mathbf{z}_{1}+\mathbf{z}_{2} ?$ Why?c. Finish the proof that $W^{\perp}$ is a subspace of $\mathbb{R}^{n}$ .

a) $c \mathbf{z} \in W^{\perp},$ and $W^{\perp}$ is closed under scalar multiplication.b) As $\mathbf{u} \in W$ is arbitrary and $\mathbf{z}_{1}+\mathbf{z}_{2} \in W^{\perp}$ follows that $W^{\perp}$ is closed under addition.c) Thus, $\mathbf{0} \in W^{\perp}$ .From all the above results, it concludes that $W^{\perp}$ is a subspace of $\mathbb{R}^{n}$ .

Calculus 3

Chapter 6

Orthogonality and Least Square

Section 1

Inner Product, Length, and Orthogonality

Vectors

Harvey Mudd College

Baylor University

University of Nottingham

Idaho State University

Lectures

02:56

In mathematics, a vector (…

06:36

02:12

Suppose $\mathbf{y}$ is or…

04:25

Let $W$ be a subspace of $…

01:53

Prove that if $\left\{\mat…

01:41

Prove that if $\mathbf{u}$…

06:26

Let $V$ be a real inner pr…

01:40

Show that $\mathbf{u} \tim…

01:18

Suppose $W$ is a subspace …

02:03

Let $\mathbf{v}$ and $\mat…

10:45

Show that the given set of…

16:09

Consider the vectors $\mat…

So we're supposing that vector y is orthogonal to both you and V So we know that. Why don't you and why Don't be or equal zero Nah, just by definition. And we want to show that why is our abominable toe every w in the span of human? Be so we have w in the span of you and B um, using the hit we can write this w has see once you plus C to me for some scaler So you want to see to So we want to show that why is also orthogonal to w So you want to show that why stocked up use equal to zero So let's try to compete wide out w So we can take why dot see you on you plus c to d So this is equal Thio why dot see you want you plus why dot c to d and by the properties of um the dot product we can pull scale er's out So we have C one at times Why don't you plus c two times wide up b and we know that Why don't you and wideout veer both zero? So we have zero plus zero which gives us here Ko So we shown that why and w oh, our fourth optimum

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…

Suppose $\mathbf{y}$ is orthogonal to $\mathbf{u}$ and $\mathbf{v} .$ Show t…

Let $W$ be a subspace of $\mathbb{R}^{n}$ with an orthogonal basis $\left\{\…

Prove that if $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k}…

Prove that if $\mathbf{u}$ is orthogonal to $\mathbf{v}$ and $\mathbf{w},$ t…

Let $V$ be a real inner product space.(a) Prove that for all $\mathbf{v}…

Show that $\mathbf{u} \times \mathbf{v}$ is orthogonal to $\mathbf{u}+\mathb…

Suppose $W$ is a subspace of $\mathbb{R}^{n}$ spanned by $n$ nonzero orthogo…

Let $\mathbf{v}$ and $\mathbf{w}$ denote two nonzero vectors. Show that the …

Show that the given set of vectors is an orthogonal set in $\mathbb{C}^{n}$,…

Consider the vectors $\mathbf{v}=(1-i, 1+2 i), \mathbf{w}=(2+i, z)$in $\…

03:50

In Exercises $9-12,$ find a unit vector in the direction of the given vector…

01:05

Find the value of each given expression.$|4-3|+|-1|$

04:44

Suppose that $\left\{\mathbf{p}_{1}, \mathbf{p}_{2}, \mathbf{p}_{3}\right\}$…

09:13

In Exercises $5-8,$ find the minimal representation of the polytope defined …

01:13

Mark each statement True or False. Justify each answer.a. If $\mathbf{y}…

04:26

Write the solution set of each inequality if x is an element of the set of i…

01:38

Two distinct points on the number line represent the numbers $a$ and $b$ . I…

16:07

Orthogonally diagonalize the matrices in Exercises $13-22,$ giving an orthog…

03:53

Show that if $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$…

00:48

Mark each statement True or False. Justify each answer.a. A set is conve…

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.