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In Exercises 27 and $28,$ view vectors in $\mathbb{R}^{n}$ as $n \times 1$ matrices. For $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^{n}$ , the matrix product $\mathbf{u}^{T} \mathbf{v}$ is a $1 \times 1$ matrix, called the scalar product, or inner product, of $\mathbf{u}$ and $\mathbf{v}$ . It is usually written as a single real number without brackets. The matrix product uv $^{T}$ is an $n \times n$ matrix, called the outer product of $\mathbf{u}$ and $\mathbf{v} .$ The products $\mathbf{u}^{T} \mathbf{v}$ and $\mathbf{u} \mathbf{v}^{T}$ will appear later in the text.If $\mathbf{u}$ and $\mathbf{v}$ are in $\mathbb{R}^{n},$ how are $\mathbf{u}^{T} \mathbf{v}$ and $\mathbf{v}^{T} \mathbf{u}$ related? How are $\mathbf{u} \mathbf{v}^{T}$ and $\mathbf{v} \mathbf{u}^{T}$ related?

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Algebra

Chapter 2

Matrix Algebra

Section 1

Matrix Operations

Introduction to Matrices

McMaster University

Baylor University

University of Michigan - Ann Arbor

Lectures

01:32

In mathematics, the absolu…

01:11

06:35

In Exercises 27 and $28,$ …

13:49

(a) Show that if $\mathbf{…

05:13

Exercises $22-26$ provide …

09:12

Let $A=\left(\begin{array}…

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In Exercises $17-20,$ show…

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The $n \times n$ matrix $\…

15:15

05:16

We study the dot product o…

02:18

02:03

Let $\mathbf{r}_{1}, \ldot…

So for this problem, we have two vectors U and V that are in our So we are treating them actually as u N v r mhm en by one dimensional nature seats. So they have en rose one caller and we want to answer How are U transpose V and V transpose You related where u transpose V is referred to as inner product And how are you ve transpose and the u transposed or transpose related where doing the UV transpose is referred to as the outer product. So if we consider our transposition rules, we didn't know that the actually, I'll start. That's one second here, so u transpose the is equal to u transpose times v transpose transposed because if we transposed twice, we get what we started, which means that u transpose V. Now what we can do here is that u transpose the transpose transposed that is the same thing as the transpose times you all transposed using the fact that if we have a transpose be, or actually if we have generally you have a b transpose, then we get be transposed a transports. So we get that u transpose V is equal to the transpose of the transpose you. But since u and V our end by one matrices u transpose v is going to be Yeah, well, I'm not going to say it's going to equal, but you transpose the is a one by one metrics, which means that obviously, since it's a one by one matrix, you know it's actually a scaler u transpose v transposed. If you transpose a scaler, you just get scaler back so similarly would get that the transpose you transposed is equal to v transposed to you. So that means that new transpose V is equal to the transpose you. The order doesn't matter then, for the outer products. Well, you ve transposed. We can't do that same trick if they're in r N u v transpose is going to give us an n by n matrix. But what we can do you can say that you ve transpose is equal to you train or you transpose transpose and v transpose, which is equal thio v transpose, um or sorry not feature exposed v times you transpose transposed. So this is the most clear relationship that we can get for the outer product, that the outer products are the transpose of each other

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