mathbfm-let-hoperatornamespanleftmathbfu_1-mathbfu_2-mathbfu_3right-and-koperatornamespanleftmathbfv

Question

$[\mathbf{M}]$ Let $H=\operatorname{Span}\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}ight\}$ and $K=\operatorname{Span}\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}ight\}$
where
$$
\begin{array}{r}{\mathbf{u}_{1}=\left[\begin{array}{r}{1} \ {2} \ {0} \ {-1}\end{array}ight], \quad \mathbf{u}_{2}=\left[\begin{array}{r}{0} \ {2} \ {-1} \ {1}\end{array}ight], \quad \mathbf{u}_{3}=\left[\begin{array}{r}{3} \ {4} \ {1} \ {-4}\end{array}ight]} \ {\mathbf{v}_{1}=\left[\begin{array}{r}{-2} \ {-2} \ {-1} \ {3}\end{array}ight], \quad \mathbf{v}_{2}=\left[\begin{array}{r}{2} \ {2} \ {-6}\end{array}ight], \quad \mathbf{v}_{3}=\left[\begin{array}{r}{-1} \ {4} \ {6} \ {-2}\end{array}ight]}\end{array}
$$
Find bases for $H, K,$ and $H+K .$ (See Exercises 33 and 34 in Section $4.1 . )$

Doruk Isik · Accepted Answer

in this problem were given six vectors. You want you to know you three at the one beat and beat and were testifying the basis for a TSH system H K and H plus. Now let&#x27;s write estimation. Reduce your for and that ISS 10 000100 to negative 100 it was You can see we have to give a Tele Vince one and one. So we have to pivot columns. It means that the person second probable or genetics will be the basis. So the basis for a church will be you want. And you too first and second hole. Let&#x27;s write the system. Can you reduce tropical? Or that is you come to 1000 0100 three negative to zero zero again. As you can see, he had to give it elements and it means that we had to give its columns. And those are the 1st 2 columns. And on the basis for a system kay, that will be the one and me too. Now I write, uh, h I wrote H plus day using exercise 34 in section 4.1 and in a reduced role. Econ form This system looks like 1000 0100 to negative. 100 1010 to negative 300 Ford, negative. 463 and zero. Now we haven&#x27;t won&#x27;t be ableto element here. First of all, we have a second favorite element here in second problem. And we have another pivot Telemann here in the orto. So our give its columns will be the 1st 1 the 2nd 1 and the portal. So it under basis for this system will be The first column is you one second billings YouTube and Fort Polo is the war.

Doruk Isik · Answer

in this problem, we&#x27;re given a set of factors and there were also given the relationship is so let&#x27;s write. Even in terms of beach Jeremy, we want to sit with three V two minus five. We three or in the same way we can find a relationship what we want internal B or B to enter guppy when I would be so that we turn will be fight me free over three minus. Sorry. Plus, we&#x27;ve gone over feet and then we can do the same for wheat free. That means free will be You went over five negative plus the, uh, Phoebe to over five. So it means that we want is a function of BTR me three and we choose a functional be the only one and b three&#x27;s a bunch of people having to. So from the 1st 1 we can say that the space for my vector B one b two b three. Well, Benny, same as this space for my electors. Be to envy. Free? No, From the 2nd 1 we can remove you to this time. So spend we want me to be brain. Believe you spend 3331 Because we too can be written in terms of weekly and we want And for literal we can say that Spend We want me to be free. We&#x27;ll be going to spend the one I&#x27;m too. Now we&#x27;re more B two b three can be retained Turns off you wanna meet? So no, we do old is Artie basis for age. So return in three Be too warm and we want we think

Doruk Isik · Answer

okay in this problem were given this system. And where s to find the numbers you&#x27;re about to be beating for section light? And we&#x27;re giving up. And what we know is the vector. And this is the question that we&#x27;re trying to Seoul. And then we&#x27;re gonna write this equation in majors form. So let&#x27;s right, everyone 53 age. Now you two or 34 down. Negative for the three. Leading to on fire before zero. Well, when he ate, Now our moves are C one C to C three and C four and right consignment. Dude says you can soup over here. It&#x27;s all here. So this is the system that we&#x27;re trying to Seoul. Now let&#x27;s find me. Let&#x27;s write this. Amen Tricks. This is our A list. Right piece magics in physical form. So we have 100 010 But it sits over three and zoo 014 Here we have si orn see to see before and again right inside Breakthroughs old here. So for the last line, we see that c three Linus or see poor is it zero? So let&#x27;s assume she poured two people. Let&#x27;s say C for inside the womb. There. Hold this wee one c three to be Oh, no. The second raw gives us any question up or C too. Waas Winnie Six or B C 40 And again we&#x27;re taking C four. It&#x27;s one so refined. See to to be negative. 26 over three and first Raw gives us any questions for C one minus C four, B zero and bonus. We find C one to be 10 over three. Now we know that the blue is open full sea or me more plus C to me too. Minus C preview tree minus before before Now we&#x27;re gonna use the seas and what we know are these. So we find tell you to be eight late to 16. Negative eight.

Michael Jacobsen · Answer

in this exercise we want to determine if this vector HK were H and K are any real numbers is in the span of you envy provide here? Let&#x27;s start off by saying that HK is in span You ve if and only if h and K as a vector is a linear combination of you envy Algebraic Lee That would be if and only if x one times you plus x two times V is equal to HK So if his victory system is consistent then we&#x27;ll say yes. HK is in the span so to determine if this victory questions consistent, we can go toe undocumented matrix where the first column is going to be you to native one calling to his V which is 21 and then we augment with the right hand side, which is HK Now. When we roll reduce we only have to go to echelon form But let me first take the falling row operation where we inter chains rose ones and two So my new row one is negative 11 k and Row two is now to to h now This way we won&#x27;t have to divide by two making this entry of one and forcing a fraction over here. So next let&#x27;s take this is our pivot entry and eliminate the entry down below. We could multiply row one by to add it to Row two, and if we copy Row one, the result would be zero four and hpe plus two K. Now let&#x27;s analyze the pivots. We have a pivot and calm one, a pivot and column too. And so it follows that this is not a pivot column and the values of H and K don&#x27;t not make that determination because of the non zero entry here. So that means that this system is consistent next one ew plus X two v equals H K is consistent for all h and K in our and so that means you envy is a linear combination. Very Excuse me, H and K is a linear combination of you envy. And so that means that HK is in the span of the vectors you envy and this holds for all values of H and K. Since our vector HK here was left in arbitrary form

Doruk Isik · Answer

being this problem, we&#x27;re given three vectors. We want me to be that close to basis be and were given a choice, which is the space spent by these records. And we&#x27;re giving it like thanks and were testifying The ordinance Spectral x. So we&#x27;re trying to Seoul a system off before it&#x27;s won. We won, plus x two me two plus extremely three fizzing of X were given X B three. Me too, and we want at Ex One Next extreme would be the corn in back to the components off the ordinance Director. Thanks. So let&#x27;s try this system. Aim matrix for Let&#x27;s Say we haven&#x27;t implemented mentions A and that is, uh, negative 649 or a three negative 3793 nated 95 native A three and we have our eggs. Electorates is for seven Negative A and B. Let&#x27;s look at the review. Straw echoed for or rented a and that is 100001000010 and Elissa, we have three bye two and zero. Well, as you can see, this part is a set. Be right. So he has three holes and he has 123 elements, actually, uh, meat dustbin age. So we show the 1st 2nd this point is actually the solution to this system. So since he has a solution, we can say that X is in age. Now, let&#x27;s find the, uh, coordinate back for X exes, then three be one plus five feature plus two, b three. And since descriptions for the corner Inspector xB from this is 352

James Chok · Answer

Okay, you this question? We want to show that the settle Victor&#x27;s B one b two b three are in fact at basis for hatch on that, actually supporting. So, what we do in this situation is we simply creating matrix where we write metrics A has bean B one, B two, B three and finally X. All we need to do is reduce this into road. Just wrote a song form on DDE. In theory, if it is in fact a basis if you want me to be three are a basis. Then we should have freed. Give the columns. So here I create apartment script Where this is the matrix, This is Matrix. Is this so the first way we&#x27;ll have minus 68 minus time. Four and second road will have full ministry five and seven and so on. So on. And so this step performs of registration form, and here we can see that the reduced for extreme form does, in fact, have 123 minutes now to find the xB coordinates we need to look at is simply this last call 352 And in fact, xB is, in fact, three. Why don&#x27;t you

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[M] Show that $\{t, \sin t, \cos 2 t, \sin t \cos…

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so $\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{v}_{1}\right\}$ is a basis for $\mathrm{H}+\mathrm{K}$

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Linear Algebra and Its Applications

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so $\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{v}_{1}\right\}$ is a basis for $\mathrm{H}+\mathrm{K}$

Linear Algebra and Its Applications

Anna Marie V.

Kayleah T.

Caleb E.

Kristen K.

Vectors Intro

Vector Basics - Example 1

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Anna Marie V.

Kayleah T.

Caleb E.

Kristen K.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications