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Determine which sets in Exercises $15-20$ are bases for $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$ . Justify each answer.$$\left[\begin{array}{r}{1} \\ {1} \\ {-2}\end{array}\right],\left[\begin{array}{r}{-5} \\ {-1} \\ {2}\end{array}\right],\left[\begin{array}{r}{7} \\ {0} \\ {-5}\end{array}\right]$$

Yes. Place the three vectors into a $3 \times 3$ matrix $A$ and determine whether $A$ is invertible:$A=\left[\begin{array}{rrr}{1} & {-5} & {7} \\ {1} & {-1} & {0} \\ {-2} & {2} & {-5}\end{array}\right]=\left[\begin{array}{rrr}{1} & {-5} & {7} \\ {0} & {4} & {-7} \\ {0} & {-8} & {9}\end{array}\right]=\left[\begin{array}{ccc}{1} & {-5} & {7} \\ {0} & {4} & {-7} \\ {0} & {0} & {-5}\end{array}\right]$The matrix $A$ has three pivots, so $A$ is invertible by the IMT and its columns form a basis for $R^{3}$

Algebra

Chapter 2

Matrix Algebra

Section 8

Subspaces of Rn

Introduction to Matrices

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to check if they're Lee. If the former busies so thought the check. If the vetters from a busies we need to check when it's a check. If the linen independence if they our linearly independence. Ah, Now do we go about checking these? So we do these by checking if the matrix formed by these vectors, that is our 11 minus two minus five minus one. Sue 705 We check if this is in vegetable because if it is inviting boot and shows that the columns are legally independent So how do we go about checking days? So we tried to make sure that we have three pie votes was after the pie votes. Then it is in vegetable. So let's rock. Let's reduce these. Who do? I found it on everything on. Then it's 10 So we do are too equal toe arsu minus are worn. And then our three or Dr Road 3/4 or all three lost suit. Times rule warned. When you do that, you're gonna off want 00 minus 54 minors. EADS seven minus seven nine. Yeah, that's where you're gonna get so next aren't on these 20 So what I do is I I do, um Or three. He called so rude. Three blows So rude, Sue. So you have won't minus 57 04 minus seven 00 minus five. So we don't even need to reduce this tree. Rachael on form. We just ask that this trip I vote. There's a pile of this pie votes So we see that this matrix as Chery Pie votes on dance is impossible and therefore says it's in vegetable. They are literally independence. I didn't have the independence that implies that's the former business.

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