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

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

Calculus 3

Chapter 4

Vector Spaces

Section 3

Linearly Independent Sets; Bases

Vectors

Campbell University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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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'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'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't won'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.

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