Linear algebra and matrix operations: correct me if im wrong

In summary, there is no such thing as matrix division in linear algebra. However, the operation of division by the right inverse of a matrix can be denoted as multiplication by the inverse. It is important to note that this is only valid when the matrix has an inverse and the number of columns is the same as the number of rows of the right inverse. In the case of a column vector, it can only have a right inverse if it is a 1x1 non-zero matrix.
  • #1
tandoorichicken
245
0
as far as intro linear algebra is concerned, there's no such thing as matrix division and its not valid to say that for a linear transformation T(x) and its standard matrix A,

[tex]A = \frac{T(\vec{x})}{\vec{x}} [/tex]

just because [itex]T(\vec{x}) = A\vec{x}[/itex]


Am I right?
 
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  • #2
It's been awhile since I've done linear algebra, but this seems like more of a semantic question than anything else. Division is simply multiplication by the inverse. (My LaTEX skills also stink, so I'll have to do this in words.) It may be more correct to say

A = T(x)*inv(x) than A = T(x)/x,

but I think you'd be in the right to say that's division. It assumes, of course, that the inverse of x exists - but if it doesn't, you couldn't divide by it anyway.

Your professor may have other opinions than mine - I'd suggest you ask her.
 
  • #3
tandoorichicken said:
as far as intro linear algebra is concerned, there's no such thing as matrix division and its not valid to say that for a linear transformation T(x) and its standard matrix A,
[tex]A = \frac{T(\vec{x})}{\vec{x}} [/tex]
just because [itex]T(\vec{x}) = A\vec{x}[/itex]
Am I right?

Actually, there IS such a thing as "matrix division" if the matrix has an inverse- multiply by the inverse matrix.

There is, however, no such thing as "vector division" which is what your second question is about. It certainly would make no sense to write what you have above.
 
  • #4
HallsofIvy said:
It certainly would make no sense to write what you have above.


Thanks. I didn't think so.
 
  • #5
If Ax = T(x), and x has a right inverse y (a matrix such that xy is the identity matrix with the same number of rows as x), then:

Axy = T(x)y
A = T(x)y

If you want to denote right multiplication by the right inverse of x as division by x, you can do so, but this only makes sense when x has a right inverse, and when the number of columns of whatever you're dividing has the same number of rows as the right inverse of x. If x is a column vector, it won't have a right inverse unless it is a 1x1 non-zero matrix.
 

FAQ: Linear algebra and matrix operations: correct me if im wrong

1. What is linear algebra and why is it important?

Linear algebra is a branch of mathematics that deals with linear equations and their representations through matrices and vector spaces. It is important because it provides a framework for solving complex problems in fields such as physics, engineering, computer science, and economics.

2. How do you solve a system of linear equations using matrix operations?

To solve a system of linear equations using matrix operations, you can use the method of Gaussian elimination or matrix inversion. These methods involve manipulating the coefficients and variables of the equations to reduce them to a simpler form, making it easier to solve for the unknown variables.

3. What is the difference between a vector and a matrix?

A vector is a one-dimensional array of numbers, while a matrix is a two-dimensional array of numbers. Vectors are used to represent quantities with magnitude and direction, while matrices are used to represent systems of equations or transformations.

4. Can you explain the concept of matrix multiplication?

Matrix multiplication is a way of combining two matrices to create a new matrix. It involves multiplying the elements of one matrix by the elements of the corresponding column in the other matrix and then summing the products. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.

5. How is linear algebra used in machine learning and data analysis?

Linear algebra is used extensively in machine learning and data analysis to represent and manipulate large datasets. Matrices are used to store and organize the data, and operations such as matrix multiplication and matrix decomposition are used to extract meaningful information from the data. Linear algebra is also used to develop algorithms for tasks such as regression, classification, and clustering.

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