exercises-9-12-display-a-matrix-a-and-an-echelon-form-of-a-find-bases-for-operatornamecol-a-and-ma-3

Question

Exercises $9-12$ display a matrix $A$ and an echelon form of $A$ . Find bases for $\operatorname{Col} A$ and $\mathrm{Nul} A,$ and then state the dimensions of these subspaces.
$$
\begin{aligned} A &=\left[\begin{array}{rrrrr}{1} & {2} & {-5} & {0} & {-1} \ {2} & {5} & {-8} & {4} & {3} \ {-3} & {-9} & {9} & {-7} & {-2} \ {3} & {10} & {-7} & {11} & {7}\end{array}ight] \ & \sim\left[\begin{array}{ccccc}{1} & {2} & {-5} & {0} & {-1} \ {0} & {1} & {2} & {4} & {5} \ {0} & {0} & {0} & {1} & {2} \ {0} & {0} & {0} & {0} & {0}\end{array}ight] \end{aligned}
$$

James Chok · Accepted Answer

again in this question, You want to find the color space and no spice all this matrix right here. So to find the column space, all we need to do is to find a pivot follows off this matrix right here at this difficult. The pivot columns are the same pivot columns as those in its road echelon formed, right? Yeah. So we have about it. Havoc, homes being these three forms. So therefore, our column Ah, calmness face is simply columns face for a He&#x27;s a span. All of these four metres. 154 minds too native four muted mine medicated wine five and finally 387 and six To find a null space off A you take its right echelon form and you want to Seoul. So if you said this is going to be you want to solve b X is it itches. So in parametric form, you want to solve 002000 minus 4100 three months. 20030 minus five is there and also rose. So then you wanna solve x one x two x three Export and x So you know in what this line tells you is x one plus two x two minus four extra plus three Export plus three x five is equal to zero. Just sold for X one you put minus two to decide. Plus fallin decide my street and my street. So this is going to be negative two x two plus by four x three minus three x four minus three x so x two There is no people calling for extra extra is just simply x two x three is going to be just simply two x four to export. Would you get from which he gets from this line right here. Now explore also had people call him. So export is just X for and X five we get from this one, which is negative X five music to zero x five is just is there So our mole space No space over a moment. Okay, Is the span off these actors? Minus 2000 And this is one, um full. 0000 Negative. 30 210 If and finally native 3000

James Chok · Answer

Okay, so we want to find the column space and null space off this matrix here to find the column space. We&#x27;re all All we need to do is to find a pivot columns off this matrix and the pivot column off this matrix is going to be the same as a pivot column off. It&#x27;s rational for now, The pivot column. All the right echelon forms are simply one, two and three thes $3. So, in the same sense, the column space is a space spanned by these vectors to the cold of space over A. He&#x27;s even to this man 11 minus two or minus two minus 101 and 551 and one To find a null space, we need to solve the equation. B X is equal to zero. What is all this? We ride up like this. You&#x27;re on the street there. Seven. There is no one wants to. So you&#x27;re you know, and nobody&#x27;s Yes. So what this implies is that x one x two eggs, three x or expired. So what is this? Begins here. Well, we know that X one is going to be if we move this to decide. This is obviously the service of this side. It&#x27;s going to be too. Plans x two plus lying times X minus line Time Victory three minus by times Export minus four times X by. Okay, on dhe X two is going to be three x three and plus 7 +65 Thanks. Bye now x three people call for X three extra is not increasing. Three x three Explore is going to be just to expire and once again, doesn&#x27;t people column for expire? So excellent is just ex wife. So you&#x27;re no space. A little space Well, a is the span Oh 20000 minus 93 100 Wanna spy 0000 Finally bars full said. And two, one who is zero here?

Gideon Idumah · Answer

in this case. Also given the midget sees a you called Sue three minus two minus five minus two minus Won&#x27;t sue 96 on then. Also, we have several minus 236 three Several. 33 9547 Andi reduced form is given to us US. Three. Sorry, the salon for Ms D&#x27;Angelo&#x27;s on story 00 minus one. Sue 00 Several for 00 0010 6310 Now, just like we did in question about 24. So you&#x27;ll recall dots the pie votes, columns, columns off. MICEX in form business for the column. Space off, eh? So what are the private columns we look at? The It&#x27;s a low form. So we see that this is a private school alone. This is a paper because I was a forcible zero following the rule. This is also one. This is a foster non zero value on. This is the first non zero of under. These doesn&#x27;t have in non zero value, So that means the pi boards the pilots columns off a. So we go back to a So we did that. This column is a private column column trees. A private school, Um, on, then columns for So this implies That&#x27;s the busies for the column. Space off, eh? Is three minus two, minus five minus two minus 1296 Andi 3733 No, for the no space to find it business for this. No species we solved. We put the actual A form in an augmented form. So of 300 zero minus won&#x27;t sue. 00 7400 001063 100000 So this is an actual on form with Turn it into your your reduced form. So what we do, is we? First of all, it&#x27;s on dysentery want. So it&#x27;s Seirawan equals over award over 30 and then would off. And then also, you want to turn this off. Since he won a cerebral soup, it quotes a road. Sue. Over. Soon we do this, we&#x27;re gonna have one minus one of the 37 over 3020 zero one sou zero. Through what? Sue zero, You know, 000 110 on then. 000000 And then next one of Tom Dysentery zero because on the biggest, just 0100 So we say we&#x27;re one quart Sue Warn Lost one of our three rules, Sue. So when we do that, we&#x27;re going off. Won&#x27;t. 03050 Is there one sue 03 over zoo zero 000110 and then lost me. 000000 Now, if you look at this, this is for Eddie in the rule reduced its uniform. So this will correspond soon. This is X one. So we have x one lost three x tree lost five over soup x five because it is x five equals zero on the X two lost two x three lost two of our two. That&#x27;s 1.5 x five equals zero. Then it&#x27;s God&#x27;s rule off x four plus X five equals zero. And then for this last tour of zero equal to zero. So the next you want to do is the general solution. So when you saw for general solution, you&#x27;re gonna of X one equals was just moving. You want to solve for excellent extra extra extra on x five saying that Casey going off X one He quotes minus three x three minus 2.5 x five then ext. You because to minus two x three minus 115 It&#x27;s five now. We don&#x27;t have an ex Terry. So just the extra because the x three x for we have that b minus its five. No, you haven&#x27;t expired. So we see it&#x27;s fine, of course, to ex fight. Just take any funding for That&#x27;s when you simplified these. You can write this US minus three minus two. Want 00 more replied by I&#x27;m just pulling out the ex Terry. So in this place, there&#x27;s no extra. So this is zero. This is zero was led by Ex Terry Loss X five. I have minus 2.5. I have minus 1.5 in the cold. One have zero because there&#x27;s no expires at zero, and this one I&#x27;ve minus one on it. I have one multiplied by X five so we can see that the business will just be his values. So there&#x27;s that busies for the no space

Sean Thrasher · Answer

to find the north orginal basis for the column space at this matrix. The column space was the subspace spanned by the column vectors of the Matrix. And so, essentially, we gotta spectra subspace That is the span of X one extra extra. And I&#x27;ve labeled them like this, right? And so I want to find a new basis for you want me to be three? That is the same. So that is the basis for the same subspace w the column space. However, I want this set of actors to be orthogonal, so v one not be too. Must be 031 dot be three. Must be zero. And so must be vey to dot B three. Okay, So clearly, we&#x27;re gonna have to apply the Gram Schmidt procedure. OK to x one. Being this vector X to being this vector next three being this vector. So what does Gram Schmidt to tell us? It&#x27;s a V one is equal to x one. Me, too is equal to x to minus a subtract x two dot v one Polar mont theme Magnitude squared of everyone. Lots of the one. And so this makes X two perpendicular to be one. And now for the three attic X three, I make it perpendicular to be one. Andi, I also make a perpendicular to V two. Okay, so this right here and is what I need to use formula that I need to use. Okay, so let&#x27;s go to the next page. So we know what? Okay. So, basically, I want to find what V one is first, then find V two and then finally find V three. Well, V one is the same as X one, as we said. So I&#x27;ve written that down V two. So let&#x27;s just focus on B two. It&#x27;s this right comes okay. We need to figure out what this V two is is a vector. What? That what? The components are, by the way, have written them like this as rose. Just a safe space. Nothing else. Okay, so, as always, you should find what these guys are First, to make things a little bit easier. So what is x two dot v one? That&#x27;s the same things. X two dot x one. Right. So it&#x27;s three plus three plus five plus five. That gives you 16. What is V one more squared? That&#x27;s one plus one plus one plus one. So that&#x27;s four. And so this is the same thing is X two minus 16 by four times V one. So that&#x27;s X two minus four times being one. So let&#x27;s figure out what V to actually is. So that&#x27;s three minus 3 to 55 minus four times one minus 1011 And so that&#x27;s the same thing is three minus four minus three plus four to five, minus 45 minus four. And so that gives me minus 11211 Let me write that down minus 1121 point. Okay, so let me update. What we&#x27;ve got so far is one minus one zero&#x27;s There, there. 11 is V one. V two is minus 11211 Now the question is, what is V three? So let&#x27;s go back to what Gram Schmidt tells us. Be three is given by this. Okay, as always, let&#x27;s figure out what these quantities are, first of all, so x three don&#x27;t be one. So that&#x27;s, uh, five, five times one minus one plus two plus eight. Okay, so five minus one plus two plus eight. So that gives me 14. Okay, x three dot ve to. That&#x27;s so be careful that you don&#x27;t use X to You need to use this guy right here. So it&#x27;s minus five plus one plus six plus one plus hates. Okay, so that&#x27;s, um Okay, so let me just check. Minus why was one plus six plus two plus states? Okay, so that&#x27;s 10 plus two. That gives me 12. The square of the magnitude of the one. Well, we&#x27;ve already worked that out. That&#x27;s just four square the magnitude of the two. So one squared, plus one squared plus two squared plus one speck plus one squared. That gives us eight. Okay, so All right. So I want to be careful not to run out of too much space. So the 36 x three minus 14. Divided by. It&#x27;s a 14 divided by four. It was the one minus 12 divided by eight times. Me too. And so that gives me X three minus seven Harms. We want minus three. House beat you. Okay, so now let me actually figure out what this vector is. So now it&#x27;s going to get a little messy, so this is going to be. So let me just write it out. So one line ist ones over one minus 11211 Yep. So this is extreme minus seven. Hearts be one minus three House. Me too. All right, this is what that is. So let&#x27;s combine this into a single vector. We get five minus seven halves, plus three halves on. Yeah, just it&#x27;s down to us arithmetic at this point, arithmetic that you should be careful not to mess up because it&#x27;s really easy to mess up arithmetic. Okay, so that&#x27;s five minus 21 plus 20 to minus five and eight, minus five. And so that gives us finally 3303 It&#x27;s a 33 033 3303 Okay. And so it&#x27;s a good idea to check that the one don&#x27;t be too equals. We wound up the three because the 2.33 cause if this is Izzy, put zero. Because if this is false, then you made a mistake somewhere. Okay, All righty. So that&#x27;s not conclusion that a North Oregon or basis for the column space is given by this. It&#x27;s driven by this collection of actors. Okay, just is a note I sometimes switch between using square braces and round braces. It really doesn&#x27;t matter. All right, so that&#x27;s it. That&#x27;s this question done. I&#x27;ll see you in the next video.

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Exercises $9-12$ display a matrix $A$ and an eche…

Problem 11 Easy Difficulty

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

Linear Algebra and Its Applications

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Catherine R.

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Michael J.

Absolute Value - Example 1

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

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

Linear Algebra and Its Applications