Understanding Matrix Transformation in 2x2 Matrices

In summary, the conversation discusses the use of matrices for transformations and their interpretation as column vectors. The identity matrix is mentioned as well as its role in matrix multiplication. The concept of dot product and rotation is also mentioned. Finally, the conversation discusses a specific case where two different column vectors are used for mapping images.
  • #1
Davio
65
0
Hiya, just a quick question regarding matrices:

The following 2 column vectors are a particular form of transformation when applied to 2x2 matrices:

(1) (0)
(0) (1)

Am I right in saying the first, takes just the x component of the matrice, and the second the y component? Thing is, To me that sounds odd because the matrice's given are just matrices, with nothing to indicate they may be vectors or whatever, they seem to be just arrays of numbers.
 
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  • #2
You're thinking too hard. [1 0; 0 1] x [a b; c d] = ? (I've delimited rows by ; )
 
  • #3
If I do what you wrote, you just get the same matrix again, but if you do it by my column vectors you get only half the matrix..
 
  • #4
The matrix M={{1,0},{0,1}} is an identity matrix. When it multiplies a matrix
N={{a,b},{c,d}} the result is M N=N. In this case you can interpret the rows of the
identity matrix as unit vectors i, j in the x and y directions respectively. Then M N takes the dot product of i and j with the columns of N, thus resulting in the original N.
 
  • #5
Ah i see, what I meant is, (1, 0 ) being one vector, which is then multiplied by M. ie. M.I, which gives just a rotation through i? I can see when my 2 column vectors are combined they equal an identity matrix, however I only want, one of the columns at any time, to see what they do to my matrix.
M={1,0}, N={{a,b},{c,d}} mn= a,b ... what actually is this called? ie. its only part of the original matrix, I think the answer given is, its a rotation of some sort, along the x axis.
 
  • #6
If I understand the question as first stated, you are asking for an
interpretation of the results of the matrix multiplication I N, where
I=
1 0
0 1

and N=

a b
c,d

The result of applying the rules of matrix multiplication is I N=N, that is why I is called
an identity matrix.

As for an interpretation of the multiplication, the dot product of the first row of I
with the first column of N gives a, and so on.

Incendentially, there is no rotation here.
 
  • #7
Davio said:
Ah i see, what I meant is, (1, 0 ) being one vector, which is then multiplied by M. ie. M.I, which gives just a rotation through i? I can see when my 2 column vectors are combined they equal an identity matrix, however I only want, one of the columns at any time, to see what they do to my matrix.
M={1,0}, N={{a,b},{c,d}} mn= a,b ... what actually is this called? ie. its only part of the original matrix, I think the answer given is, its a rotation of some sort, along the x axis.

MN is not a valid matrix product, unless you meant MTN. NM is also a valid matrix product. The map f(N) = NM just selects the first column of the matrix N, the image of the first basis vector. The map f(N) = MTN maps N to the image of the first basis vector of the dual space in the dual of the image of N. If N is a rotation matrix, you will note that the result is a rotation of the basis vector.
 
Last edited:
  • #8
jimvoit said:
If I understand the question as first stated, you are asking for an
interpretation of the results of the matrix multiplication I N, where
I=
1 0
0 1

and N=

a b
c,d

The result of applying the rules of matrix multiplication is I N=N, that is why I is called
an identity matrix.

As for an interpretation of the multiplication, the dot product of the first row of I
with the first column of N gives a, and so on.

Incendentially, there is no rotation here.

AH, nope sorry, I wasn't being clear, I mean, I have 2 different column vectors, (which happen to look just liike an identity matrix when put together)
Slider142 's explanation is kinda of what I'm looking for, except I don't quite understand it.
I did mean:(1 ). M. By mapping image, do you mean, the vector is placed along 1 ?
(0 ) 0
 

1. What is a matrix transformation?

A matrix transformation is a mathematical operation in which a matrix (a rectangular array of numbers) is applied to a set of coordinates, resulting in a new set of coordinates. It is commonly used in computer graphics to manipulate the position, size, and shape of objects.

2. How is a matrix transformation represented?

A matrix transformation is typically represented by a 3x3 or 4x4 matrix, depending on the type of transformation. Each element in the matrix represents a different factor that affects the coordinates of the object being transformed.

3. What are the different types of matrix transformations?

There are several types of matrix transformations, including translation (shifting an object's position), rotation (changing an object's orientation), scaling (changing an object's size), and shearing (distorting an object). Combinations of these transformations can also be used to achieve more complex effects.

4. What are some real-world applications of matrix transformations?

Matrix transformations have a wide range of applications in fields such as computer graphics, robotics, and machine learning. They are used to create 3D animations, simulate movement of robotic arms, and process and analyze data in machine learning algorithms.

5. How are matrix transformations related to linear algebra?

Matrix transformations are a fundamental concept in linear algebra, which is the study of vector spaces and linear transformations. They are used to represent and perform operations on vectors, and can be used to solve systems of linear equations. In fact, matrix transformations are a powerful tool for solving a wide range of mathematical problems in various fields.

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