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15. Find the vector space spanned by the columns of the matrix below2 2A =~1 0 2...

Question

15. Find the vector space spanned by the columns of the matrix below2 2A =~1 0 2

15. Find the vector space spanned by the columns of the matrix below 2 2 A = ~1 0 2



Answers

If $\mathbf{a}_{1}=\left[\begin{array}{lll}1 & 2\end{array}\right], \mathbf{a}_{2}=\left[\begin{array}{ll}3 & 4\end{array}\right],$ and $\mathbf{a}_{3}=\left[\begin{array}{ll}5 & 1\end{array}\right],$ write the matrix $$A=\left[\begin{array}{l} \mathbf{a}_{1} \\ \mathbf{a}_{2} \\ \mathbf{a}_{3} \end{array}\right]$$ and determine the column vectors of $A$

Were asked to find the null space of a definition of the no space is given at the top here, with the no space of an MBA and matrix is given by the set of vectors eggs and are in such that the matrix, a times the Vector X is equal to the zero vector. The problem gives the Matrix and he has a three by three matrix shown here. We can see from there that value of N is three, which means we're in or three so we can write our expect er as a three by one matrix with X one next to the next three. So now we can start the soul of this linear equation, but using reduction to get a in a row echelon form. So every Retin A and augmented form down here we can start our row reduction by seeing that we can get, we can get a zero here If we multiply and minus four times the first row and add that to the second room, we're gonna add what is four. I'm drew one and added to that gives our first rule stay the same. 11 is 21 and uh, zero. So I have a minus four times one, and it's a four, which gives zero. We have a minus four times a minus to plus and minus seven, which gives warm. We have minus four times one plus of minus two, which is minus six. Their third road stays the scene minus 134 Then we have a minus four times zero, which is zero in the zero of the third room. So now we can We can get a zero here if we add one. Times were one to row three. So add one times were one the road three. So that will give our first rebels stay the same. Their second rule stay the same on our third row will be one plus and minus one, which is zero. Then we have ah minus two minus to plus three, which is one. And then we have one A C one plus four, which is five. Then a 000 Now we can get let's see weaken, multiply the second row by minus one and that the three and that will give us a zero here, several multiple it or we will we will end minus one one's road to and and that the three So our first row stays the same or a second room will stay the same. 01 minus 60 Then our third rule have a minus one times zero, which is zero minus one minus one times one Edits the 10 the minus one times six plus five which is 11 the most one time 00 which is zero. Now we can dio one more step. Uh, to get a one year, we can multiply that third row but one ever 11 for most playing the third row by one every living We'll show that down here So our first true will stay the same one minus 210 2nd row stays the same and zero minus one minus six zero in our third room will be zero zero and 1 11 times a left. 11 is one and zero. So that's our Rochel Inform Matrix A And now we can multiply that by or expect Cherie because x one x two x three that is equal to zero factory which is a three by one matrix was here s so that we can multiply this out that will have first row x one minus two times x two and a one times X three is equal to zero. Next, we have a zero times x one minus one times x two the minus six times x three 00 and then we have zero times x 10 times X two and one times x three is equal to zero so we can see that x three equals zero. If we played that in here than X two has two equal to Z equals zero. Flex two is equal to zero. This was an ex too. So if x two and x 3/2 the equals zero, the next one has to equal zero that are expected has one solution which is, well, zeros. So we can write the solution to our problem that the North space of a because one solution which is zero factory. That's the answer

We were asked to find the null space of a The mission of the null space is given up top here, where the null space oven n by n Matrix A is a set of vectors x and are in such that the matrix a times the Doctor X is equal to the zero. The growing gives us the Matrix A is a three by three matrix shown here We can see that in equals. Three unions were in or three, which means our expect er's we're gonna be three by one virtues x one thanks to the next three. So now we can start solving this linear equation by reducing the matrix a tow reduced through a national informed. So we'll start that road, bro. Reduction blue. I've re written matrix A and I meant it form so we can start our row reduction and we can first move this term to zero. If we multiply a negative three fiber one and add that through to so writing that shorthand, we will multiply negative three by row one and added the road to that will give us minus three times one. So I'll rewrite our row. One that won't change. We have minus three times one plus three, which is zero minus three times I plus four I, which leaves us with all right and minus three times minus two plus or minus five. Which gives us one and my three times Europe list 00 in a row. Three states the same when it's one. When is three I I and zero. So now we can think this terms of zero. If we multiply row one about one and then just add that 03 and so we can write that shorthand is multiplying room one by one and adding it to three. So we have where one stays the same for two stays the same. Then we have adding row. 123 We have one minus 10 I minus three eyes minus two I and they in the minus to Plus I is I minus two in zero plus zero zero. Now we can we can multiply the second row, but to than that and that's a road three. Metal. Get rid of this term so we will multiply, wrote two by two and add the the three I was scrolling down. We have first Roach. There's the same one I minus two zero and multiplying right to by two. We have to time zero, which for third rule to time zero plus zero zero. Two times I plus a minus two I zero. So all right. At this second room, which there is the same I 10 then a to times one plus an I minus two leaves us with just and two time zero plus here is still zero. Now we can multiple I wrote to, but to and then and that's a real one. And then we can get rid of this term here, this term here. So we will be multiplying two by road to and ending it through one. So all right, out our third rooms, your zero i zero her second round will stay the same. 010 Now we're multiplying our second row by two and adding at the ones that we have to time, zero plus one is one hands a two times I plus eyes three, uh, in the two times it two times one plus a minus 20 And we have to time zero plus 00 And then we can now we can multiply Row three I I to get rid of the complex number. And also we can do the same thing with room to do so we'll do. Two things in the same operation will multiply Row three I I and multiply row to buy I So our first road stays the same for a second row is our time 005 times I which is minus one and hard times one, which is I and I think 00 when multiplying or three by I'm We have our time 00 I time 00 times eyes minus one at time 00 So we've reduced this to Rochel Informant weaken Soul We could start We can multiply our major cities together and sold this so rewrite are reduced matrix so are reduced matrix a and we're doing the a Times X which is equal to the zero factor. So we have our use matrix a one three I zero and then our second room zero minus one and I in our third row 00 minus one. Well, that times are expected, which is x one x two x three is equal to or zero Victor which is a three by 1000 so we can multiply this out one times x one three times x two and a zero times X three is equal to zero. We have a zero times x one minus, one times x two, which is minor sex, too. Plus five times X three equals zero. Then we have a zero times x 10 times x two The A minus one times x three sub minus x three is equal to zero. Well x three has to equal to zero has to be equal to zero. So that zero, which means X two has two equal zero, which means that X one has to equal zero. So we have one solution here, which is the zero victory so we can write our solution as the null space of a. So we have our extractor here, which we have one solution. So that means the null space of a is the zero victor. That's the answer

Were asked to find a null space of a and the definition of the null space is given at the top here with the null space oven. N by N Matrix A is equal to the set of vectors X in or in such that the Matrix, a times the Vector X is equal to the zero vector. The problem is matrix A as a three by two matrix, with the first column being to one minus three in the second column being minus forward to my Spi we can see from are a matrix that are in value is equal to two, which means we're in or two, Which means our expect er is gonna be a two by one matrix with X one and X two. Now we can go ahead and solve this mystery equation. Um, then we'll do that by putting a in Rochel informed by using road reduction. So we'll start with the reduction and I have the Matrix a re written down here. So this is the matrix. Every rate rewritten and augmented form of zeros on the right hand side. And to reduce this into a relational, inform the first thing we can do is per mutate Row two with row one So and I can abbreviate that with a P for perm et row one into and that will give the new Matrix 12 through minus forward and minus three minus five zero is still on the right. No, the next thing we can do, we can We can get a zero in the bottom left hand corner. If we add three times row one to that third red making abbreviate that within a for ad, we're gonna add three times row one to row three. So that will give us. Everyone will still be seen for a two will still be the same. Now we're adding three times were ones that were three times one plus minus three, which is zero. Let me have three times to, which is six plus minus five, which gives us one and our zero's air. Still in the right hand side, we have three times zero plus zero is zero Now the next thing we can do is we can get zero on the first row are on the first column of our second row. If we add minus two times, red ones are zero to. So you know, abbreviate that is adding minus two times were one. So to that'll give are saving through one. Let me have minus two times one plus two, which is zero. Then we have minus two times two, which is minus four plus minus four, which is minus eight. And our third row stays the same and or zero stay the same. Now this matrix is reduced enough to where we can sold it. I'll rewrite it down Here, begin. We're solving linear equation. The Matrix. Eight times. The X is equal to 00 vector. And so we have our reduced me reduced matrix A as 120 minus AIDS and 01 And we have our expect. Er with excellent next to that's equal to zero. No, that looks like 10. Or rewrite that. Yeah, that's equal to director. So multiplying this l we get x one plus two x two equals zero. We have a zero times x will in which zero minus and eight times x two is equal to zero. We have a zero times x one, which is zero and a one times x Two years of X two equals zero we have this x two equals zero. And if we pulling that into our second equation, that still just zero equals zero. It looked like that appears X two equals zero. We get that X one is also you go to zero so x one equals zero. The next two x two equals zero and that is one solution. So are no space contains only contains the zero factory for eggs is just zero bacteria. It's a two by one matrix. Double zeros. Now we can go ahead and write our solution out for the middle space for a solution will be that the null space abbreviated by capital in in the no space of a is equal to zero vector. This is the answer.

Okay. The problem is asking us to write a matrix and, um in the second part of it is to find the column vectors after the right to make. So this is very simple because we have and equal to 81 18 83. We already given to us today. One 83. Siphoned right down. Wanted to three and four five. And lots of dads are matrix and the columns are two columns. 135 That's one comet tree Director. This is too 41 That's the second column factor. So we wrote our Matrix and the two column vectors. Thank you.


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