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$[\mathbf{M}]$ In Exercises $37-40,$ let $T$ be the linear transformation whosestandard matrix is given. In Exercises 37 and $38,$ decide if $T$ is a one-to-one mapping. In Exercises 39 and $40,$ decide if $T$ maps $\mathbb{R}^{5}$ onto $\mathbb{R}^{5} .$ Justify your answers. $$\left[\begin{array}{rrrr}{-5} & {10} & {-5} & {4} \\ {8} & {3} & {-4} & {7} \\ {4} & {-9} & {5} & {-3} \\ {-3} & {-2} & {5} & {4}\end{array}\right]$$

by the above result, the transformation $T$ is not one-to-one.

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 9

The Matrix of a Linear Transformation

Introduction to Matrices

Missouri State University

Campbell University

Oregon State University

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we have to find if the given transformation, um is 1 to 1, but not, um in order to find out the kind of cancer we have to use the road reduction that we know already and then looking at the rank and looking at the number off pivot columns pivot positions in each column. We can decide if it is wonder one or not. So let's start by, You know, we will keep the first. The role always fixed. Thank you. We will try to the malls on the streets here that understanding a screen. So if the first ro is fixed, I'm going to fix it at the same bad deuce. What? I'm going to change. But I am going to do the following task first. Odd to what will be my new second row? Well, it will be the previous second room. Added bit eight. Bye, five times album. Okay, so how How much will that be? Let's see. See, first IHS, we have to multiply eight by five with the entire first. True. So what will be it if I multiply by eight by five, this will become minus eight, right? This will remain zero This will become minus eight on this will become, um, four into eight by five. So it will be 32 boys. Thanks. Correct. So I'm showing it for one. And then I will expect that you would be able to follow for the the rest of the rules 100. So now our took That means what? Previously my art to first position was eight. So eight plus minus eight. How much we like it. We will get here. Cedo. Same way three plus zero. I'm sorry. It's not zero. Still, so 10 will then become If I write 10 over here, See, it will be 16th. Correct. I have a 16 over here. So my first well was 33 plus 16. It will give us 19. Then I have again minus eight plus minus four. So it will be minus 12 and then seven. Plus, Did you do by five? So seven less 32 divided by five. It will be 67 because 75 sir. 35 35 plus 32 is 67. So we get 60 seven. Bye. Fine, right. Same way now. The next action that I'm going to do is are three are three will be the older three. Plus are one is going to be changed. So whatever numerator off art that will come in the denominated divided very five. And whatever is the coefficient off are three that will go in there New Morita. So four by five, Hard one, Correct. So again, if I clean up the screen little bit I'm sorry. Then we will see what will be those numbers. And as I again say that for one particular But I'm showing all the steps I hope you would be able to figure out for the remaining glands. So now if I multiplied with four by five this number four by five I will change it to, uh, minus. I'm sorry when you take up minus for if I multiply this number with four by five, I will change it toe plus off eight. If I change this number, multiply this number with four by five I will change it to minus four and four by five. So 16 by fine Now what am I doing? I am adding this toe The previous told you so. Let us add element by element four plus minus four. It will yield zero again. Minus nine plus eight. It will yield minus one plus five minus four. It will yield plus one and then minus three plus 16 by five. So it will build one bite. Fine. Next the same way. If you see on four, he's going Toby at Fort plus the coefficient off. Three. Bye. Coefficient off first. So playing well, it one. So definitely the first term will be zero. The second term would be minus eight. The third term would be eight and the last time would be eight by five. Now, if you see these two rules 00 if you C minus one left, minus seat minus 11195 minus 88855 It's like there is an eight right throughout. So why can't we subtract the fourth rule from the first one after we multiplied the first one? Right? So let us try that. I am not writing the first two rows. Imagine There, there, in the third one, we will have this and my new four through will be our for prime, which will be eight times third row minus only for truth. So that will be zero my insider. Whatever multiply you get zero the eighth time for a third row. So it will minus it minus the next troop. So it will be zero. It will be zero. It will be zero. So now, at this point, if we take a look at the system, this is giving of a metrics that somehow looks like this minus five 10 minus 54 0 19 to a minus 12 67.5 zero 19 minus 12. 67 by five. Zero minus 115 I mean, one by five. Seattle minus one one one by five and then zero zero zero Syria. Now, what is so particular about it that it is the moment if it position food. Hello? Okay, look. Awesome, Trump. This means that this system off recreation have a free baby. This is means nontrivial solution on a dark to me. We dad? Mm, Yes, please. A little nice Mobile insulation. Mhm. It's more than one position means the consternation. P confirmation. Qin's the transformation t don't be. Come on, transformation. So that's a That's a visit

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