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Determine which sets in Exercises $15-20$ are bases for $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$ . Justify each answer.$$\left[\begin{array}{r}{3} \\ {-8} \\ {1}\end{array}\right],\left[\begin{array}{r}{6} \\ {2} \\ {-5}\end{array}\right]$$

No.

Algebra

Chapter 2

Matrix Algebra

Section 8

Subspaces of Rn

Introduction to Matrices

Campbell University

Oregon State University

Baylor University

University of Michigan - Ann Arbor

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Okay. Good day, ladies and gentlemen. Um, today we're looking at problem number 19 from section 2.8. Onda question asked us. Eso were given these two factors u and V on it wants to know whether or not u and V form a basis for our three. Um, actually, technically, that's not even what the question asked. It says, does u and V form a basis for R two or R three? Um, okay, uh, and in both cases, the answer actually is No. So let's let's explain why. Why? So is, um, Han. Or maybe I should say is um, you the A basis or are to Okay. And the answer is No, No. Why not? Well, because U and V are not in our to because, um, you and the are not not in our too, because our two is two dimensional vector. Um and that would mean that it would be a vector of the form X y eso It would have to coordinates, but U and V are three coordinate vectors. So, um, you, uh, you come of er three coordinate vectors three or three core or three vectors? I guess you could call them three vectors or three coordinate vectors. Um, U and V are in our three u v n are three. So e mean to be short about it. There are three coordinate vectors. Okay, They have three coordinates, so they have to They're not in Our two there have to be in our three. Okay. Eso eso next and so not in our to, um Then we have to consider our three. So, um, ex question is is, um u v a basis for far three. Okay, so now, based on what I said first, U and V are in fact, in our three, So you the are in our three. That is true. So they are three dimensional vectors because they have a new X and X coordinate a y coordinate and z coordinate. So they are in our our three. But eyes I claim to you, they're not a basis for, but they are not a basis. Um, you and the not, uh, a basis for our three. And now why would this be true? Well, the reason it's true is because, um, to be a pieces for our three means you need to span the whole space. So, in other words, um, it needs to be a three dimensional subspace off our three. Uh, in other words, um, the sun space, uh, spanned I, u and V must be three dimensional, three dimensionally from three dimensional. Um, okay, Okay. I did say you and veer three dimensional vectors, but there's only two factors. Um, but but, um, you the are on Lee to our are on Lee. Two vectors, two vectors. So what I mean is that there's only two factors there. So that means, um, you ve can spanned a space can on Lee. The largest space, maybe I put it this way. The largest, um, subspace off are three spanned. Bye. U and V, Um uh, is two dimensional is the largest possible. Maybe I say it this way. The largest possible, um, subspace spanned subspace of our three span by U and V, um is two dimensional is two dimensional, because there's only two factors. The subspace, um, the dimension of the subspace band is that most the number of actors in the set. So in this particular now, we have two vectors, so that means the dimension of the subspace could be at most two is the largest possible subspace of our three span. By is dimension as to dimension is, um, as two dimensions. In fact, in this case, I believe it would end up being, um, a two dimensional space. But anyhow, as two dimensions. So that means, um, it cannot be. Are three hence, hence, um oops. Ah, you the not a basis or, um, are three. So our conclusion is that it can't be a basis for our three because you only have two factors since you only have two vectors. Um, doctors cannot span a three dimensional space, and this is by definition, it can. The definition of a three dimensional space is in one sense, um, that any spanning set has toe have three vectors. Three literally. Independently. So, um, so that's it for this problem, folks. Thank you very much And have a nice day

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