Linear algebra-how do I know if something is invertible?

• frasifrasi
In summary, to determine if a linear transformation or function is invertible, you can find its matrix representation and check if the matrix is invertible. For reflections about a line, you can use the vertical line test for functions or the horizontal line test for inverse functions. Additionally, any reflection is its own inverse.
frasifrasi
Linear algebra--how do I know if something is invertible?

Say I have a "reflection about the line y = x/3 in R^2," how do I know if the function or "T" is invertible?

Thank you.

T is inv if its matrix is invertible, I think reflecting about y = x/3 function/T is invertible .

just use vertical/horizontal line test for functions

Well you could first find out what the matrix representation of this linear transformation is. Are you familiar with these terms?

yes, but how does the vertical line test work?

I don't think it would work on transformations >>

for functions draw a vertical line, if it passes through the function twice then that means your function is not an actual function

and for inverse, draw horizontal line ...

something really simple stupid that I never used

Well there is an invertible transformation for any reflection across a line through the origin.

Geometrically, if you reflect a second time across the same line, you are right back where you started. Any reflection is its own inverse.

What is an invertible matrix?

An invertible matrix is a square matrix that has an inverse, meaning it can be multiplied by another matrix to produce the identity matrix. In other words, the inverse of an invertible matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix.

How do I know if a matrix is invertible?

A matrix is invertible if its determinant is non-zero. The determinant is a scalar value that can be calculated from the elements of a square matrix. If the determinant is equal to 0, then the matrix is not invertible. However, if the determinant is non-zero, the matrix is invertible.

What is the importance of invertible matrices in linear algebra?

Invertible matrices are important in linear algebra because they allow for the solving of systems of linear equations. They also have many other applications in fields such as physics, engineering, and computer science.

How do I find the inverse of a matrix?

The inverse of a matrix can be found using various methods, such as Gaussian elimination, Cramer's rule, and the adjugate matrix method. These methods involve manipulating the original matrix in a specific way to obtain the inverse matrix.

Can a non-square matrix be invertible?

No, a non-square matrix cannot be invertible. In order for a matrix to have an inverse, it must be a square matrix, meaning it has an equal number of rows and columns. Non-square matrices do not have a determinant and therefore cannot have an inverse.

• Calculus and Beyond Homework Help
Replies
5
Views
364
• Calculus and Beyond Homework Help
Replies
4
Views
977
• Calculus and Beyond Homework Help
Replies
1
Views
679
• Calculus and Beyond Homework Help
Replies
6
Views
1K
• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
488
• Calculus and Beyond Homework Help
Replies
8
Views
814
• Calculus and Beyond Homework Help
Replies
11
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
614
• Calculus and Beyond Homework Help
Replies
1
Views
1K