Matrices

Agent Smith

Agent Smith

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Oct 18, 2023
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What's the relationship between

\(\displaystyle A = \begin{bmatrix}1 & 0\end{bmatrix}\)

and

\(\displaystyle B = \begin{bmatrix}0 & 1\end{bmatrix}\)

???
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\(\displaystyle \begin{bmatrix}1 & 0\end{bmatrix} \times \begin{bmatrix}a \\ b\end{bmatrix} = \begin{bmatrix}a \end{bmatrix}\)

\(\displaystyle \begin{bmatrix}0 & 1 \end{bmatrix} \times \begin{bmatrix}a \\ b \end{bmatrix} = \begin {bmatrix} b \end{bmatrix}\)
---

Visually, A and B are mirror images of each other. What does that mean in matrix math? A simple test done shows that, for a \(\displaystyle 2 \times 1\) matrix, A selects the \(\displaystyle x_{11}\) element and B selects the \(\displaystyle x_{21}\) term.
 
This is a pretty vague question. Are you asking because it is a part of a larger problem/question?
 
A and B are mirror images of each other reflected in the line y=x.
Are you asking what this 2x2 transformation matrix is?
 
Look carefully at what happens when you multiply by the identity matrix. Just don't say that you get back the same matrix.
 
Harry_the_cat said:
A and B are mirror images of each other reflected in the line y=x.
Are you asking what this 2x2 transformation matrix is?
Click to expand...
One can also treat them rotated versions of each other. Any matrix of of the family below will transform A to B:
[math]\begin{pmatrix}0 & u\\1 & v\end{pmatrix} A = B[/math]
 
blamocur said:
This is a pretty vague question. Are you asking because it is a part of a larger problem/question?
Click to expand...
Humblest apologies ... I never got to study matrices well. Firstly, it wasn't a big part of the high school curriculum and secondly, it's way to abstract for me.
 
Harry_the_cat said:
A and B are mirror images of each other reflected in the line y=x.
Are you asking what this 2x2 transformation matrix is?
Click to expand...
A and B are [imath]1 \times 2[/imath] matrix and I was wondering how I could transform A to B and vice versa.

[imath]T \times A = B. \therefore T = ?[/imath],
 
blamocur said:
One can also treat them rotated versions of each other. Any matrix of of the family below will transform A to B:
[math]\begin{pmatrix}0 & u\\1 & v\end{pmatrix} A = B[/math]
Click to expand...
Si, this is what I'm aiming to do. Vide supra.
 
blamocur said:
What dimensions do you expect [imath]X[/imath] to have?
Click to expand...
Dimensions should be kept to a minimum.

Dimensions(A) = Dimensions(B) = 1 × 2
So Dimensions(T) = 1 × 1. The reason is from how matrix multiplication is defined. I don't think such a matrix exists, oui? Je ne sais pas.
 
That matrix will have to be 2x2.
The matrix you need is

1720410077768.png
but you will need to write A and B as column matrices instead of row matrices.
 
Agent Smith said:
just trying
Click to expand...
I was hoping for a 'yes' or 'no' answer, but, okay. My motivation for asking is three-fold. You'd posted a question that had already been answered. A substantial number of your prior posts in the forum do not seem serious, to me. Yet, if you are serious about learning basic linear algebra, may I suggest one of the free, introductory video courses available online? This way, you will progressively learn definitions, vocabulary, basic forms, and uses, and you can post your questions in a progressive manner without members needing to cover basics in order to answer them. Of course, people may answer most of your questions, regardless, but trying to self-teach broad mathematical topics through some kind of flirtation with the topic generally wastes energy.
[imath]\;[/imath]
 
Harry_the_cat said:
That matrix will have to be 2x2.
The matrix you need is

View attachment 38305
but you will need to write A and B as column matrices instead of row matrices.
Click to expand...
Vectors, the convention (if no rationale exists) is to have them as row/column matrices?

An indeed, it seems I picked the wrong dimensions for my matrices. Most helpful.
Gracias.
 
Last edited:
mmm4444bot said:
I was hoping for a 'yes' or 'no' answer, but, okay. My motivation for asking is three-fold. You'd posted a question that had already been answered. A substantial number of your prior posts in the forum do not seem serious, to me. Yet, if you are serious about learning basic linear algebra, may I suggest one of the free, introductory video courses available online? This way, you will progressively learn definitions, vocabulary, basic forms, and uses, and you can post your questions in a progressive manner without members needing to cover basics in order to answer them. Of course, people may answer most of your questions, regardless, but trying to self-teach broad mathematical topics through some kind of flirtation with the topic generally wastes energy.
[imath]\;[/imath]
Click to expand...
Sound advice!

Gracias.

The question is, veritas, basic: the only 2 elements in my matrices are 1 and 0 (for the moment). Apologies if this annoys anyone; mathematics is not a walk in the park and I struggle.
 
You can form a 2x2 linear transformation matrix in the following way:
1. What is the image of the point (1, 0)? Write this as the first column in a 2x2 matrix.
2. What is the image of the point (0, 1)? Write this as the second column.
Done!
 
Harry_the_cat said:
You can form a 2x2 linear transformation matrix in the following way:
1. What is the image of the point (1, 0)? Write this as the first column in a 2x2 matrix.
2. What is the image of the point (0, 1)? Write this as the second column.
Done!
Click to expand...
Most gracious of you to extend a helping hand. I forgot we could combine stuff into the same matrix by increasing its dimensions. Will check that out, ASAP.

How is the vector [imath]a\hat {i} + b \hat {j}[/imath] usually represented? [imath]\begin{bmatrix}a & b \end{bmatrix}[/imath] or [imath]\begin{bmatrix}a \\ b \end{bmatrix}[/imath]?
 
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