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The row echelon for of matrix A is given by: 2 0Then a basis for the Nullspace of A is:~2102the above setthe above SetHHFk Vn4bo /"Nonf Ouingae...

Question

The row echelon for of matrix A is given by: 2 0Then a basis for the Nullspace of A is:~2102the above setthe above SetHHFk Vn4bo /"Nonf Ouingae

The row echelon for of matrix A is given by: 2 0 Then a basis for the Nullspace of A is: ~2 10 2 the above set the above Set HHFk Vn4bo /" Nonf Ouingae



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A matrix in row echelon form is given. By inspection, find a basis for the row space and for the column space of that matrix. (a) $\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{array}\right]$ (b) $\left[\begin{array}{cccc}1 & -3 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{array}\right]$

All right. Okay. Um so, given a matrix in row echelon form. Okay, so which looks like the following? Okay, 5310. These are Zeros, apologize for my uh poor handwriting on the tablet. Okay, so this note that this is a five by four uh matrix. So, uh if you just look at the roads long form, right? You see that one of the roses all zeros. Okay, So that means that if you look at the dimensions of the road space in the column space of this matrix, which we will call a for now. Okay, So the dimension of both the road space and the columns, column space is going to be four. Right? Because if you look at the columns, okay, there are four uh columns, none of which is all zeros. So that's fine. To the dimension of the column spaces for and for the road space there are 44 rows in which uh none of the entries are all, you know, None of the roads are all not all zero. Okay. And there's one row where it's all zeros. Okay, So five minus one gives you four. Okay? So in terms of finding the basis for these, all you have to do is to eyeball. It's okay. So for the road space because this is given in a row echelon form, we can just Go through the rose one x 1 and that would be the basis. So 1245 01 -30 001 -3 and 0001. Okay, so this would be our basis. Okay, for the road space for the columns based it's only slightly uh more difficult. All you have to do is to read through the columns. Okay, One by one. So it's 100021000 And then for negative 3100 And then last but not least. 50 negative 310 Okay, so these four vectors form the basis for the column space of this matrix. So let's look at another example. So suppose we have the following four by four uh 2100, and 5 3 -71. Okay, so again uh if you inspect the this matrix which is in row echelon form, you see that none of the rose nor the columns are all zeros. Therefore the road space and the column space are both going to be a full dimension for. Okay. So what does that mean? That means that we can just read off the rows and the columns of this matrix and get our basis for the road space and column space. So in particular for the road space, we just read it off. So it's 12 negative 15 01, 4, 3. Uh huh. 001 negative seven and 0001 Okay, so that's the basis for a row space. And for the column space, we just read off the column. So it's 10002100 negative 140 and 53 negative 71 Okay, so this is the basis for a road space.

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'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

Okay, so we want to find the column space and null space off this matrix here to find the column space. We'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'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'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're on the street there. Seven. There is no one wants to. So you're you know, and nobody'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'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't people column for expire? So excellent is just ex wife. So you'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?


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