💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Stuck on your homework problem? This step-by-step video should help.

Try Numerade Free for 30 Days

Like

Report

Let $\mathbf{p}_{1}=\left[\begin{array}{r}{2} \\ {-3} \\ {1} \\ {2}\end{array}\right], \quad \mathbf{p}_{2}=\left[\begin{array}{r}{1} \\ {2} \\ {-1} \\ {3}\end{array}\right], \quad \mathbf{n}_{1}=\left[\begin{array}{l}{1} \\ {2} \\ {4} \\ {2}\end{array}\right], \quad$ and $\mathbf{n}_{2}=\left[\begin{array}{l}{2} \\ {3} \\ {1} \\ {5}\end{array}\right] ;$ let $H_{1}$ be the hyperplane in $\mathbb{R}^{4}$ through $\mathbf{p}_{1}$ with normal $\mathbf{n}_{1} ;$ and let $H_{2}$ be the hyperplane through $\mathbf{p}_{2}$ with normal $\mathbf{n}_{2} .$ Give an explicit description of $H_{1} \cap H_{2}$ . [Hint: Find a point $\mathbf{p}$ in $H_{1} \cap H_{2}$ and two linearly independent vectors $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ that span a subspace parallel to the $2-$ dimensional flat $H_{1} \cap H_{2} . ]$

Then $H_{1} \cap H_{2}=\left\{x : x=p+x_{3} v_{1}+x_{4} v_{2}\right\}, x_{3}, x_{4}$ are free variables.

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 4

Hyperplanes

Vectors

Johns Hopkins University

Missouri State University

Oregon State University

Idaho State University

Lectures

02:56

In mathematics, a vector (…

06:36

03:51

Let $\mathbf{a}_{1}=\left[…

$[\mathbf{M}]$ Let $H=\ope…

02:43

In Exercises 11 and $12,$ …

In Exercises 1 and $2,$ yo…

06:08

05:00

In Exercises $7-10$ , let …

01:47

Let $S$ be the subspace of…

02:18

Let $\mathbf{y}, \mathbf{v…

09:46

Consider the vectors$$…

01:41

Okay, so in this question, we want to find the hype we want to find. There exists a higher things which separates a one and beat so the heart abstain in this question, who all is given by three x one plus by X two months by two x three. Now we have to compete with distances for a one, and all of them is three times to give you six minus one minus. Two times five gives it 10. So that is this minus five f bulb a tuba Music three times three Giving mine plus one minus. Two times three Getting one of six. So that's four f o a. Three is it 23 times minus one using negative three plus six. So that's just three Now, if you will be one is equal to three times a week. Zero plus five minus two times minus one gives a prostitute That's seven. It will be true. Is it? Two? Three times one gives you three Meyers three minus two times by my studio plus four. So that's just four o. B. Three is equal to 23 times to deep six plus to minus two times 2 to 1. So that's six. So now you want to find a high plane who separates these values altogether, Separates this values and these. So if we draw one about High Plains where offenses even sued What f off X, he's a zero. So we just have this FX. So this thing is, if you 20 what you see is that out? How a one will lie a one lie How a one Spain, We'll lie above here. Well, like yeah. Oh, so a one state will live with this one. So if oh, explore gifted minds five lap that. But then everything else will, like low like a lotus plane, A rosy nose. So then we haven't separated a ones and the ones so instead, to separate them what we'll do these will set effort. This effort is equal to four in this case, in this case, this plane minus five plane. Well, that's just that was three plane that's re sign. We'll lie somewhere here it's the bigger bubble fo x Ah, he's even 23 and the modest five plane like just appear above that. Uh, if X by is into my file and then these tangle, of course. Libel, Otis. So just for one of them, okay. And so f o X bar is excuse six every moment on seven. So, yes, we found a high plane which can separate these two. But the problem is this point right here at eight. To and be to lie on this high thing. So there is not high plane that's strictly strip separate A and B, so the answer is no.

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…

Let $\mathbf{a}_{1}=\left[\begin{array}{r}{2} \\ {-1} \\ {5}\end{array}\righ…

$[\mathbf{M}]$ Let $H=\operatorname{Span}\left\{\mathbf{v}_{1}, \mathbf{v}_{…

In Exercises 11 and $12,$ find the closest point to $y$ in the subspace $W$ …

In Exercises 1 and $2,$ you may assume that $\left\{\mathbf{u}_{1}, \ldots, …

In Exercises 11 and $12,$ find the dimension of the subspace spanned by the …

In Exercises $7-10$ , let $H$ be the hyperplane through the listed points. (…

Let $S$ be the subspace of $\mathbb{R}^{3}$ spanned by the vectors $\mathbf{…

Let $\mathbf{y}, \mathbf{v}_{1},$ and $\mathbf{v}_{2}$ be as in Exercise $12…

Consider the vectors$$A_{1}=\left[\begin{array}{rl}1 & 2 \\<…

Let $S$ be the subspace of $\mathbb{R}^{4}$ consisting of all vectors of the…

09:12

Construct a spectral decomposition of $A$ from Example 3

00:53

Find the singular values of the matrices.$\left[\begin{array}{rr}{1} &am…

02:06

Let $Q(\mathbf{x})=7 x_{1}^{2}+x_{2}^{2}+7 x_{3}^{2}-8 x_{1} x_{2}-4 x_{1} x…

03:44

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

07:43

Let $A$ be the matrix of the quadratic form$$9 x_{1}^{2}+7 x_{2}^{2}+11 …

04:21

Determine which of the matrices in Exercises $7-12$ are orthogonal. If ortho…

02:07

Repeat Exercise 7 for the data in Exercise 2

02:39

In Exercises $3-6,$ find (a) the maximum value of $Q(x)$ subject to the cons…

02:12

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

03:40

Repeat Exercise 23 for $\mathbf{v}_{1}=\left[\begin{array}{l}{1} \\ {2}\end{…

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.