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IL IVH F 1 L Ji 1 [ 1 1 1 3 2 2 | il 8 11 8 1 L Wi 1 1 8 a 1 I 1 : 8 1 1 L...

Question

IL IVH F 1 L Ji 1 [ 1 1 1 3 2 2 | il 8 11 8 1 L Wi 1 1 8 a 1 I 1 : 8 1 1 L

IL IVH F 1 L Ji 1 [ 1 1 1 3 2 2 | il 8 11 8 1 L Wi 1 1 8 a 1 I 1 : 8 1 1 L



Answers

$\left[ \begin{array}{lll}{1} & {1} & {1} \\ {1} & {2} & {3} \\ {0} & {1} & {1}\end{array}\right]$

So in this problem, we're given this matrix and were asked to use the matrix capability of a graphing utility to find the determinant. So I went to Desmond's dot com, went up here to math tools, matrix calculator, and got this matrix calculator here. So in our new matrix and our matrix is a four by four. So four rose and four columns. Now the entries one minus one, C A A and four and two and six zero my S four and then to zero two and six and then zero two, eight zero. Okay, where's my matrix now? Just do D E T right here to get the determinant determinant of a yes, Negative,

This video's gonna go through the answer to question number 11 from chapter 9.3. So ask to use real reduction to find the inverse off the matrix. That 11 one 121 Thio three. So So we conform the combination matrix with the identity and they tried refugees. Okay, so if we subtract to you off the top equation from the bomb equation, then we're gonna get zero one that to you, minus 20 Maybe it's gonna be minus 201 on the inside. And if we should bottle subtract one of the first question from the middle equation, that's gonna be zero That's gonna be one on that's going to zero months. Well, on zero on me, the top equation as it is, Savior zero. Okay, so now we get to be a stick in court because on left inside the bomb equation on the middle or after the bottom row of the majors in the middle of the matrix. All the same, which means that the ah, the row is off the matrix linearly dependence, which by their a born in the book, means that er the identity that's all right with me

They're. So for this exercise we have this vector B. And the subspace dovey generated by the one, V two and V three that are these vectors that are defined here. So basically we need to calculate the Earth a little projection of you on this space to view. And just remember remember this projection is calculated as the inner proud of the vector V. Each of the generators of this subspace dog. In this case the generators RV one, The two and 3. So we need to calculate the we need to calculate the inner part of me with each of the generator divided the score of the norm of the generators times degenerates. So these for the three vectors B two square plus the interpreter of B would be three. B three. Did the square of the norm of B. Three. Okay, so just to remind you a little bit of the geometric intuition of this, is that the view is generated by these three vectors. So what we're doing is projecting we on each of the generators and then some that together. So we want We t. v. one and V three acts as a basis. Actually in this case they are linearly independent so they form a basis for this. Yeah, subspace of you. So we're writing the in terms of this basis. So we're projecting projecting on this sub space. So let's calculate the correspondent values that we need. So in this case we would be one. The product of B would be to dinner product of the would be three. So this is equal two, one half, There is a constitute and this inner product is equal to zero and then the norms. So because this is the cost to zero means that we don't need this term anymore is going to be equal to zero. So we just need to calculate the score of the norms for B. two and B one. So for me, one square of the norm, remember that there is equal to the inner product of the vector with itself. And in this case this result in one and the inner approach of B two square is equal 2, 1 as well. So these are actually military vectors. And then we just need to put all together on the four. So behalf that the projection of the vector B on the subspace, our view, it's equals to 1/4 times 11 one plus the vector V two. That is equal to one, 1 -1 -1. After some. In these two vectors obtain the action solution that is one half times the vector, three, three minus one minus one. That corresponds to their thermal projection of beyond this subspace of you.

And one more matrix modification here. This time of the times a Roman. eight times I Okay. We will run through what how this works. Okay, a resultant two by two. Matrix again, we're just following our matrix multiplication rules. Row one, column one. One times one. The Stereo Times four will give us a one. Okay, top right. That row one, column two, one times two, zero times three. That will give us a two there. Mhm. Caught him. Right. Row two column oil one time zero. That's four times 1 for their and then I lost out on the bottom right, We know his road to call him too. Zero times two. It was one times 3 for three. His matrix looks familiar. Again, it is a matrix. So multiplying the identity on either side here, either on the left or on the right will not change the matrix that that identity is multiplied.


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