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Use the ________________ member function to remove the last element from a vector....

Question

Use the ________________ member function to remove the last element from a vector.

Use the ________________ member function to remove the last element from a vector.



Answers

Find a unit vector in the direction of the given vector.
$$\mathbf{v}=\langle 1,-1\rangle$$

In this example, we have a linear transformation. T that maps are three back to our three. It's going to be defined by t of X equals eight times X, where here are matrix. A is defined now for this transformation. Let's also let B B the vector set equal to negative 17 Negative three. Now the questions are for what? X in our three is t of X equal to be so let's determine what the solution X is in this equation now. T of X recall is equal to eight times X, so the question could be rephrased As for what X is a of X equal to be. But as soon as we write our question this form, we can get the augmented matrix and begin row reducing to solve for X. So let's first take the Matrix A. It's one negative to three 01 negative, too, and negative to six negative five then augments with the Vector B, which is given to be negative 17 and negative three. So now that we form the Matrix, the next step is to begin row reduction. So I'll copy Row one and our objective is to obtain the role reduced echelon form. So for the pivot one obtained here, we need to eliminate all entries above and below. So in this case, we need to eliminate to negative two and three. So let's multiply Row one by to add it to row two and we will get 012 and a five. Next, if we multiply row one by three or negative three, rather at two. Row three will obtain zero negative too. Ah, positive one and a two. So now we have a pivot here with zeros below. For this pivot, we have zeroes zero above, but not below. That means on the next step, we're going to be needing to eliminate that value. Negative too. So let me copy Rose one and two. Next. The first row is 10 negative to negative one. And our new row two is your A 1 to 5. Now, to eliminate the negative to multiply row, to buy positive too. And add to row three will obtain zero zero ah, five and looks like there's a tiny mistake here. This should have been a zero. And that means this entry here should be equal to value of 10. So now let's check our pivots at this stage here in here, we have to pivot positions with zeros above and below. And our next pivot position will be this highlighted. Five. We need to make the 501 So for the next row operation divide Row three by five we'll obtain 0012 and let me copy the rest. So we have 0125 and 10 negative to negative one. Now, on the next step, from this pivot position we see here, we need to eliminate the entry to and negative to found above. So this time let's start by copying row three, which is your 012 Then multiply row three by negative to add the results to row two, we'll obtain 01 zero and then a one for the next operation. Multiply Row three by positive to this time and added to row one will attain 100 three. Now, at this stage, we have the identity matrix here augmented with values on the right, and this tells us that X one must be equal to three. X two is one and x three is too So here's our conclusion. Then if we set a vector X to be equal to 312 then the image of X under T is equal to the vector B, and this solves this problem.

So we want to find about your ex such that a X equals B. Um, or this is our matrix A. And this is our vector B. So we want to row reduce the augmented matrix. So first, um, I want to make these two value zero using the first equation. So first and third equation are gonna remain the same. So now let's change the second equation. I'm gonna take negative three times the first equation and add it to the second. So we get a zero here. Negative three times negative to a six. Minus four. Gives us too negative three times wanted negative. Three plus five is, too. And then we have negative three times. One, which is negative. Three plus nine gives us six. And we're gonna do a similar thing for the last equation. Take three times the first equation and add it to the fourth. So one times 33 minus 30 Uh, times three is minus six plus five is minus. One times there is three. Minus four is minus one on time. Serious, three minus six is negative. Three. So now our first columns taken care of. Um, next, we have In the second rule, we have 02 to 6. So we can, um, cut everything in half. That's one of the rules were allowed to do with, uh, producing matrices. So instead, I'm gonna write 0113 So now I'm going to use this new row two thio, get rid of these two values. So I'm gonna multiply this new row to buy negative one and add it to this room surgery here. So I have a zero. I have zero, um, times negative one. And at one, a zero three times naked of Juan is negative. Three plus three is zero. And then I'm gonna do the same thing for the last equation. Um, I'm gonna multiply this row two we have in our matrix here. Bye. Positive one this time and add it to the Lord. So that makes this negative one a zero. Um, again makes a negative on a zero. And ah, we have three postnegative three, which is zero. So since we have rows of zeros, um, we can expect to have a free variable, So let's see. Um, what these equations say? First equation says X minus two. Why? Plus z equals one second equation says why plus C equals three, and that's all we have. So the second equation let's rewrite it as why equals three minus C and substituted in the first equation. So we have X minus, too. Why so over not replace that with three minus C plus Z equals one facility, right? X minus six plus two Z plus Z equals one. So finally we have X plus, they're easy equals seven. So, uh, one of these variables has to be free, so it's juicy to be free. Ah, so our general solution iss going to be our vector X iss equal to seven minus three z three minus c access the expression we had for why and Z is free so we can rewrite this as, um 730 plus some vector times are free variable. So first equation has negative three z Certain put minus three. Here. Second has a negative one z. So negative one and last is one time. See? So here's our expression for our General Vector X. If we want a particular solution, um, we can pick Zee to be some regular value s. So let's just take the equals one. So if Z equals one, um, we just have 730 plus negative three negative 11 which gives us 4 to 1, so that's one exit would work.

Okay, so four problem for two. We need to consider the Matrix that we did before, and and we want to. I want to find a column of this matrix that it can be doing. Did it end? Have the remaining matrix columns deal? Spend our four. So instead of kind of considered the original Mitrice, I would recommend to just to consider the real reduced form off the previous matrix, which is I'll just write again. 10 connective Seven over 13 and 00 zero juan 01 connective to over 13 and 00 and 0001 zero Andi 00001 So if we count the people columns, this is one and this is another one. And here's to the odd one. And this the other one. So the calm that we can to delete in this case is Oh, you're not a color. Is this column because this is another people economies. If we delete this column, then that will not affect hourly new independence off these four pictures. So that means call him. Three can deleted so that these poor rector's steel for this four columns were still span are for. And another question is that can delete wanted one calm, because Thea answer is no. Because we have. For we only have four people columns. And if we wait, we have five columns in total way. Have deleted one column that it's not people, columns and any any more columns. If we deleted, that will be people columns. And that will not be that will not be allowed in this case. Because if we delete one when people calm said we only have three people comes that means Sorry. Uh, we don't have three people of columns. Three people call. So you, Stephanie not enough cannot cannot spend our are for But you can't spend our three but not possible to spend our four

This problem Got to find in on zero Vitto in the north. Off, eh? On in non zero of that saw in the column off a So let's do for the known off a first not to do this. We solved for eggs team a X quarts zero. And how do we do this? We do these by solving the documented many trees. E come on. Zero. So in this case, we off. 14 minus five soup. So five minus 17 370 11 The odd 0000 So we try and turn this into a reduced matrix. So what do we do? We force a vote on everything yet? Zero. Because this is a pipe one stone decency in private school. Um, so we have That's world Sue equal to road Sue minus for room Juan. Rule three equal to row three plus five for one under one roof full. The quarter for minus minus Our soo. Rule one. When you do this, you're gonna off want 000 Sue minus 393 three minus five 15 of five. I'm gonna have 0000 No, the next one of these one ton dysentery one and thereafter, everything before bm above and below it zero to make This are also a private's. So all we do is we foster for sit at the road soup equal to rude sue divided by minus three. We do that. We're going off. 1230 015 over. 30 09 15 0 0350 You know, lastly want It's on everything above this pie votes to zero and zero. So I would do is we see for a war on it Calls or warn Minus tsu road Sue Earl three equals over three minus nine or two on then Earl four equals or four minus Terry root roll three. So minus three road sue. So when you do this, we're gonna have umm one zero minus one over 30015 over. 300000 0000 Now, with these, we can solve this and this implies That's recorder. This is X one. This is extremely so that's the implies. X one minus one over a three x three equals zero. This will be ex too lost five over three ex Terry in court zero. And then this is zero quote zero. Now, when you saw for this, you're gonna have that. The general solution. The general solution will be x one equal to one of our three x three ext. Suit equal to minus five over a very extremely. Now we don't have an ex, Terry, so you can take extra to anything. So we choose would choose any value for X three. See extra Because the one say ex Terry equal to one. So this implies that the value in the note and non zero vet so would be so off. One of the three extremely said I'll be one of the three minus five over three. And then I chose this to be one. So this is wants a D's is in the north off because you can actually write this as ext. Three quartz Eck story. I can ride these ours one of the three minus five of a theory and then one more supplied by ex Terry. So this Viktor is the victor in the north space off, eh? Now, lastly for the column space off A. We have that so far, the column space off eh? We have by definition of column space that any column any cologne in a is in non zero of Epsom in column space off, eh? So you can choose any column so I can't speak. For example, I can pick the first column, which is 14 minus five suit.


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