define-f-mathbbr-rightarrow-mathbbr-by-fxm-xb-a-show-that-f-is-a-linear-transformation-when-b0-b-fin

Question

Define $f : \mathbb{R} ightarrow \mathbb{R}$ by $f(x)=m x+b$
a. Show that $f$ is a linear transformation when $b=0$ .
b. Find a property of a linear transformation that is violated when $b 
eq 0$ .
c. Why is $f$ called a linear function?

Ashley High · Accepted Answer

this question wants to know if f of X equals X plus two divided by two is a linear function. Well, it is. All we have to do is we&#x27;re gonna separate this too, for both of this exodus to appear. So it&#x27;ll look like f of X equals X over two plus two over too. So this is a safety guest saying X over to us in the ass, saying 1/2 X and then two divided by two is just one. So we can see now that this is in the form of Y equals in X plus B or 1/2 is our slope and one is our y intercept. So, yes, this is a linear function.

Stephanie Gaston · Answer

so this problem way are asked to rearrange this given function into this function. So our first step to do that would be to take our functions are starting function. I think I&#x27;m gonna multiply both sides times X minus A so e x minus and the denominator on the right. We can&#x27;t slept. And I&#x27;m also going to distribute that c so I will get see times X minus C times a equals f of X minus f of a and already we can see that it&#x27;s much similar to what we&#x27;re aiming for right there. Okay. One thing I noticed is that ffx is alone on the left hand side. So I&#x27;m gonna move my f of a to the other side so I will get Sea of X minus C of a plus is a equals effort X. So I can easily write that in the other direction equality is symmetric. And now to make it match that final thing, I really just need to rearrange the order. This is f of X People&#x27;s Seve X plus f k minus c a day or see time a roast forgetting parentheses. I&#x27;m going to add parentheses, which does not change anything. Really? So now we can see that this is in fact, a line specifically just matches for y equals MX plus B format for C does my slope And today line of C a day is my y intercept. Be so that is a line work. F is linear if you go for

Ankit Gupta · Answer

for the mill equation. If fact is equal, he makes plus, you have to find four x one plus extra. So it is nothing like him off X one plus extra plus B letters. He makes one close a mixture is the first one second wishing they are asking was for meaningful fixed one. It was a false extra. So for fix one is nothing but a mix one place and if affects Lewis and makes two people because nothing. When. M x one plus x two plus two The third question IHS It&#x27;s a question with stating that affects one IHS, fourth x one less extreme is equal to a four x one. So this is not equal. We can clearly see because he b when b is nautical zero, it is not equal. When B is equal to zero, they are equipped

Runpeng Li · Answer

We need to show that, as is a linear transformation and were given the condition that, um, t and s satisfy the credibility creations and tease and enough inspiration. Okay. How? Just write down our goal here. You near transformation. Okay. All right. He ordered to do that. We take arbitrary, take any you and B from our in. So we pluck. Are you gonna be to transferred to the ingenious information as we will have X? And why accident. Why, Okay, now we consider t off acts. Now, remember, acts can be represented as ass off you. So plugging as self you now, since ass is it, it&#x27;s the number. As an S and T said it&#x27;s by the embers in credibility questions. So this is this keeps us directly back to you. So similarly, we have, um, t apply is factor VII. Okay, so now we consider ask you plus. Yeah. So since you plus V can be represented as a transformation of x and y under T, So that will be t ext plus t y. Now remember the inner transformation by our assumption, the transformation transformation T is a leaner transformation so we can apply the property. Help me near says Formacion here at his t x s t y t X plus y okay. And by our convertibility equations, these keeps us express What? And recall that X is s you And why is SP so that ask you Class sp and our first condition It said his buy. So the next condition we need to check it&#x27;s whether s off the escape, their off vector under the interview and then you&#x27;re under the transpiration Information off ass will be the same as the skater times times the transformation off s so that is directly to show that c o s u because s self see you as I see you, that&#x27;s it. And to do that first consider considered TLC X because he&#x27;s a linear transformation. So it&#x27;s just a tee off a sea of t x c times t x and T acts by our previous by our previous reasoning here, T X will be you. It&#x27;s C times. You okay? So s off. See, Time&#x27;s view can be written as as soft tee time. See x right, which is a substitute c times you by. I don&#x27;t trust for Michel t so That&#x27;s, uh, tee times tfc times eight times X. Now we can we can apply the convertibility equations, so this gives us see X. Now, remember, ax by over by this equation here, acts can be represented as ask you so at a same as C times Ask you so as I see you is C times as here. So we&#x27;re done. That means ass, he said.

Michael Jacobsen · Answer

in this exercise were given a set of linear dependent vectors in R N Let&#x27;s start out by writing just what it means for this set of vectors to be linearly dependent, it means is that what this means is there exists scale. Er&#x27;s x one. Let&#x27;s go with C one rather see to and C three, which are not all zero such that the following linear combination is satisfied. Where we take C one times V one plus C two times V two plus C three times V three and this will be equal to the zero Vector where we said not all of these vectors air Zero. And that means that this vector equation has a nontrivial solution. Let&#x27;s inject a new element into this situation. Let&#x27;s define T, which will map from the vector space are in into RM. So that t is linear here. The most important statement about T is that this transformation is linear. So now we wonder about that are set of vectors that could be the images of tea that would be specifically TV one TV too, and TV three we know that the set of vectors v one v two v three is already linearly dependent. So the question becomes, What about this set? Should it be linearly, independent, linearly dependent? Or do we have not enough information to tell? Well, one way to determine the outcome here, using our transformation T is to start back up at this equation here. Let&#x27;s apply the transformation t on the left hand side. So that&#x27;s tea of C one times V one plus C two times V two plus C three times V three. Then on the right hand side will use the fact that since T is linear t of the zero Vector is still the zero vector so we can write in the zero back there here. Well, this equation so far, it&#x27;s giving us closer to determining if this set is linear, independent or dependent. But we haven&#x27;t gone further enough. Let&#x27;s begin using more properties of linearity. Linearity allows us to write this left hand side here in the form of C one times T a. V one plus C two times TV, too, plus C three times t at V three and this is set equal to zero vector. Sylvie in your itty allows us to break apart the vector addition as well as the scaler multiplication so that we get to this new equation. Now let&#x27;s recall what we said about C one, C two and C three. We said that they are not all zero, and here they are in this equation. So that means that this vector equation has a non trivial solution. And it&#x27;s also a homogeneous vector equation because the right hand side is equal to the zero vector. Now see what it means for a set to be linearly dependent. This set is linearly dependent now because we have a nontrivial solution to this equation, and so we can now state our conclusion TV one TV, too, and TV three as a set is linearly dependent. So the punchline to this example is that linearly dependent sets are mapped into linearly. Independent sets provided our transformation t is linear now, sadly, we might want to extend this. What if the set was linearly independent? Well, tragically linearly. Independent sets could still be mapped to a linearly dependent sets. So the statement here that this set is linearly dependent on Lee works for dependence

Problem

An affine transformation $T : \mathbb{R}^{n} \rig…

Problem 29 Easy Difficulty

Define $f : \mathbb{R} \rightarrow \mathbb{R}$ by $f(x)=m x+b$
a. Show that $f$ is a linear transformation when $b=0$ .
b. Find a property of a linear transformation that is violated when $b \neq 0$ .
c. Why is $f$ called a linear function?

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Grace H.

Anna Marie V.

Heather Z.

Alayna H.

Absolute Value - Example 1

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Linear Algebra and Its Applications

Grace H.

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Absolute Value - Example 1

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David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Grace H.

Anna Marie V.

Heather Z.

Alayna H.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications