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I really need help with this question, I do not understand matrices. 

 

Let \(\mathbf{u}\) and \(\mathbf{v}\) be vectors such that \(\|\mathbf{u}\| = 3\) and \(\|\mathbf{v}\| = 2\) such that the angle between \(\mathbf{u} \) and \(\mathbf{v}\) when placed tail to tail is \(60^\circ\).

Let \(\mathbf{A}\) be a matrix such that

 

\(\mathbf{row}_1(\mathbf{A}) = \mathbf{u}, \mathbf{row}_2(\mathbf{A}) = \mathbf{v}.\)


Then what is \(\mathbf{A} \mathbf{u}, \mathbf{A} \mathbf{v}\) in that order? 

 

 

Thanks in advance!

 Jul 18, 2022
 #1
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\(\mathbf{A} \mathbf{u} = \begin{pmatrix} 6 \\ 9 \end{pmatrix} \quad \mathbf{A} \mathbf{v} = \begin{pmatrix} 8 \\ 4 \end{pmatrix}\)

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 Jul 18, 2022
 #2
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Hi Guest!
Ok, this question is about matrix multiplicaton and the dot product (Also called: Inner product or scalar product).

u is a vector, so it has components: \(u_1,u_2,...u_n\)

Similarly, v is a vector, so it has components: \(v_1,v_2,...,v_n\)

Now, no need to actually make it in n-dimension. I mean we can, but why not simplify this and assume we are in 2 dimensional space?
That is, n=2
So, \(= \)  and v \(= \)

So our matrix A is:  \(A=\begin{bmatrix} u_1 && u_2 \\ v_1 && v_2 \end{bmatrix}\)  (As given, u is the first row and v is the second row)

But we are given an angle and lengths of these vectors. Can we find the "dot product"? 
Yes: \(u \dot {} v=u_1v_1+u_2v_2=cos(\theta)*\left || u |\right |*\left || v| \right |=cos(60)*3*2=3\) 

So, we got: \(u_1v_1+u_2v_2=3\)

Next, let's see what the question really wants:
\(Au=\begin{bmatrix} u_1 && u_2 \\ v_1 && v_2 \end{bmatrix}\) \(*\begin{bmatrix} u_1 \\ u_2 \end{bmatrix}\) = \(\begin{pmatrix} u_1^2+u_2^2\\ u_1v_1+u_2v_2 \end{pmatrix}\) = \(\begin{pmatrix} 3^2\\ 3 \end{pmatrix}=\begin{pmatrix} 9\\ 3 \end{pmatrix}\)

Now, in a similar way, find \(Av\).

Hope this helps!

 Jul 19, 2022
 #3
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I got Av= (9;3)

 

Can anyone please confirm that?

Guest Jul 20, 2022
 #4
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Edit: Sorry I meant Au= 3;9

Guest Jul 20, 2022
 #5
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The dot product of u and v is equal to ||u||* ||v||*cos 60 =  3. (Prove this before you start using it!)

row_1(Au) = \(\mathbf{u \cdot u}\) = 9

row_2(Au) = Dot product of u and v, equaling 3


Au = <9, 3>

 

Similarly:

 

row_1(Av) = 3

row_2(Av) = v \cdot v = 4

 

Av = <3, 4>

 Jul 21, 2022
 #6
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There seems to be a notational or transcription error in the question.

The statement that the first row of the matrix A is the vector u implies that u is a row vector.

(Similarly, the implication is that v is a row vector.)

We are not given the dimension of u and v, but assuming that it is two, A will be a 2 by 2 matrix.

In that case, the products Au and Av do not exist, for the products to exist u and v would need to be column vectors.

The question would make sense if we were given that

\(\displaystyle \text{row}_{1}(A)=u^{\text{T}},\)

(and similarly for v),

or, calculate

\(\displaystyle \text{A}u^{\text{T}},\)

(If u is a row vector.)

 Jul 21, 2022

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