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

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

Show that the orthogonal projection of a vector $y$ onto a line $L$ through the origin in $\mathbb{R}^{2}$ does not depend on the choice of the nonzero $\mathbf{u}$ in $L$ used in the formula for $\hat{\mathbf{y}} .$ To do this, suppose $\mathbf{y}$ and $\mathbf{u}$ are given and $\hat{\mathbf{y}}$ has been computed by formula $(2)$ in this section. Replace $\mathbf{u}$ in that formula by $c \mathbf{u},$ where $c$ is an unspecified nonzero scalar. Show that the new formula gives the same $\hat{\mathbf{y}} .$

So \hat{y does not depend on the choice of nonzero u in L used in the formula. }

Calculus 3

Chapter 6

Orthogonality and Least Square

Section 2

Orthogonal Sets

Vectors

Missouri State University

Oregon State University

University of Nottingham

Idaho State University

Lectures

02:56

In mathematics, a vector (…

06:36

02:12

Suppose $\mathbf{y}$ is or…

01:41

Prove that if $\mathbf{u}$…

02:13

Show that $\mathbf{v} \tim…

01:40

Show that $\mathbf{u} \tim…

01:53

Prove that if $\left\{\mat…

06:49

Show that the given set of…

10:45

02:14

Recall that two lines $y=m…

16:09

Consider the vectors $\mat…

01:20

Let $\left\{\mathbf{v}_{1}…

Okay, This question wants us to prove that if we're projecting a vector onto a line, the answer is independent of what vector along the line we choose. So we'll say that our line l is equal to the span of some vector, which we're gonna call you. And we're also going to say that why is the factor that will be projecting? So if this was our Y vector and this was our l vector, this red line right here would be the projection, and we can actually find a formula for the projection. So the projection on tha does vector you of why is equal to why dotted with you all over the magnitude of you squared times you. And now we're going to show that no matter what multiple of you we pick that spans the line, we will get the same answer. So the projection onto you of why would be equal to why dotted with see you over the magnitude of see you quantity squared times See you. Because again, we're just taking a scaler multiple so we can account for any vector that we could possibly shoes. So I'm gonna pull a C squared out in front. So we get C squared times. Why dotted with you over Well, the magnitude of see you quantity squared which will deal with in a minute times you and what's the magnitude of the vector? See you. Well, the magnitude of see you would just be our original magnitude scaled by sea. So by a similar argument, we just have to square, See if we're squaring the magnitude of see you. And now we see that the C Square's cancel giving us why dotted with you over the magnitude of you squared times you. And that's exactly the projection onto you of Why? So we're done.

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…

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

Show that $\mathbf{v} \times \mathbf{u}$ is orthogonal to $(\mathbf{u} \cdot…

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

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

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

Recall that two lines $y=m x+b$ and $y=n x+c$ are orthogonal provided $m n=-…

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

Let $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\}$ be an orthogonal set of …

01:55

Let $\mathbf{y}=\left[\begin{array}{r}{-3} \\ {9}\end{array}\right]$ and $\m…

09:31

Let $T : \mathbb{P}_{2} \rightarrow \mathbb{P}_{3}$ be the transformation th…

02:07

For the matrices in Exercises $15-17$ , list the eigenvalues, repeated accor…

01:08

Determine which of the matrices in Exercises $1-6$ are symmetric. $$…

03:01

Classify the quadratic forms in Exercises $9-18 .$ Then make a change of var…

01:46

Suppose $\mu$ is an eigenvalue of the $B$ in Exercise $15,$ and that $\mathb…

02:21

Find the equation $y=\beta_{0}+\beta_{1} x$ of the least-squares line that b…

01:18

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

04:58

Let $\mathbf{y}=\left[\begin{array}{l}{7} \\ {9}\end{array}\right], \mathbf{…

In Exercises 19 and $20,$ find $(a)$ the largest eigenvalue and $(b)$ the ei…

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.