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Find a QR factorization of the matrix in Exercise 11

$R = Q ^ { T } A = \left[ \begin{array} { c c c } { \sqrt { 5 } } & { - \sqrt { 5 } } & { 4 \sqrt { 5 } } \\ { 0 } & { 6 } & { - 2 } \\ { 0 } & { 0 } & { 4 } \end{array} \right]$

Calculus 3

Chapter 6

Orthogonality and Least Square

Section 4

The Gram–Schmidt Process

Vectors

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in problem 15. We want toe. Make a Q R factory ization for these metrics. The first system is to apply Graham ish minted process. So get orthogonal basis for the columns in this video. That's a ploy. Gram Schmidt, We get the one as the first victor on minus one minus 111 And we get V two as the projection as the second victor for the second column 214 minus four to and we subtract the projection for the second victor in tow, V one on minus one minus 111 We multiply by a factor to make this projection projection. This factor has in denominator doesn't product between these two victims. You have to minus one minus four. Minus four was two equals minus five. Divided by the product off. We want bond itself, which gives life equals two plus 13 one one plus two one plus one equals two. You have one minus minus minus is one minus one equal 00 one minus one equal zero. We have full minus minus one equals three minus minus minus equals minus. Then we have minus four minus my minus minus. She's plus once four plus one equals minus three. And finally we have two plus one equals three. This is V two. Then we get with three. As the third column in the metrics, which is serve. The column is five minus four, minus three five, minus four minus 371 minus the projection off this victor onto V two, 30333 Want to blow it by a factor? This factor had in denominator little product between these two. Work toe These two Victor's We have five multiplied by three minus three multiplied by three. We have minus three, multiplied by seven plus City gives minus 12, divided by little product. Between Vito and itself, we have three squared multiplied by four, which is 36 minus the projection off the third column into the one one minus one minus 111 But the blood boy a factor. This factor had in the nominator the product between the third column and the one which is for you, plus four plus three plus seven plus one. She gives 20 divided by the product off the one and itself. She gives the flavor. These equals five minus. They have minus and minus one third. Then we have plus five plus three equals it. And we have here a minus on. This gives the four minus four. We have five plus three minus four. It gives the four. Then we have minus four plus three plus one. This is not for sorry. We have here five plus three, but applied by one third, which means five plus one minus four equals two. Then we have minus four plus four, which equals zero. We have minus city minus three multiplied by answer which she is minus one. Then we have minus minus four, which is plus four, which is zero have minus three plus one plus four equals two. And finally we have seven minus three, which is full. We have seven minus minus three, multiplied by onset, which is seven plus one minus four seven minus one plus four equals two and we have one minus plus one minus four equals minus two. This is the three. The second step is to normal, Iet these victors get in one as the normalization off. We won. This is we won. We divide by. It's normal. It's normal. Is the square root of five square root off. Five. We have one divided by a squared five multiplied by one minus one minus 111 Then we get to the same way we normalize V two. This is V two. We divided by square root off 36 which is six. Then we have one divided by half the blow it by 101 minus 11 Finally, we get the normalization off the three, which is here we divided by its normal, which is the square root of 20 which is one which is to but the blogger Square toe five. Then we have here Photo by 1011 minus one. Then we can construct the Q metrics, which equals in the first column. In the metrics we have in one we can we have one. Whatever school to five minus one. Whatever. School to five minus one, divided by square to five. One divided by square 25 and we have one by the by school to five. The second column is in tow, huh? Zero, huh? Minus half off. Third column is N three have one divided by square to 501 divided by square to five one Bible school five minus one divided by square to five. The final step is to get or, sir, the step is the first part of the question off the answer off The question stuck on the step. The fourth step to get our, which equals Q transpose multiplied by a matrix C. It equals one divided by square to five minus one, divided by square to five. It's wonderful. It was great to 5151 but it was good to five off zero off, minus half half on the viable square to 501 The Bible Scholar to five. One, divided by square to five minus one, divided by square to five. We multiply it by the metrics A, which is one minus one minus 111 21 four minus four to five, minus four. When a three 71 we can make the multiplication here toe get or or equals. The first element comes from multiplying. The first rule with the first victim, which is five. The valuable square toe five equals square toe. Five. The second. The second element here comes from the the first group of the boy by the second victim, we have tu minus one minus four minus four plus two equals minus five. The web squared five equals minus square to five. And then we must blow it with Victor. We have five plus four plus three plus seven plus one for the 20 divided by a squared five equals four or the blow by square to five. Then we take the second rope on. Then multiply by the first victor with second victory and third, we have one plus zero minus one minus one. Last one divided by two equals zero, Which is very logic because, or is a triangle of metrics. Then we have two plus zero plus four last four plus two, 12 divided boy to equal six and we have five plus zero minus three minus seven plus one equals minus four by the boy two equals minus two. Finally, we take the third row and multiply by the three victors. You have one minus warm. 10 minus one plus one minus one equals zero. Then we have to below zero plus four minus four minus two equals zero, Which is very correct. Finally we have for you plus zero minus three plus seven minus swamp equals it. Divided by the square root of five It divided by square root. Five we have here a little mystic. This is not it. The boy was called five. Because we have here this note this victor and three off Incorrect because we have calculated the norm. Wrong is square root off 16, which gives four. This means we have here half Oh, zero off off a minus huff. And this we have here off zero half off a minus off which gives us it divided by two which gives four here. And this is the answer. Mhm a second barred off the answer with solution and Q are here is the final answer of our problem.

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