# Inverse of a Linear Transformation

• KataKoniK
In summary, T is the linear transformation that takes [x, y]^t to [2x, y]^t, and its inverse, if it exists, would take [2x, y]^t to [x, y]^t.
KataKoniK
Hi,

Is there a formula to do this? The textbook just says to "reverse" the action of T to get T^-1 (T inverse). Can someone explain to me in laymen terms, how to accomplish this? For example,

For T = [2x y]^T is T^-1 = [-2x y]^T?

T may not be invertible (this T isn't) inverses only exiist for square matrices, adn not all of those. you may want to look up the adjugate matrix, or any basic linear algebra textbook that'll talk about this when it discusses gaussian elimination.

I have a hunch that KataKoniK meant that T is the linear transformation that takes [x, y]^t to [2x, y]^t. Its inverse, if it exists, would take [2x, y]^t to [x, y]^t.

To make my question more clear, in the following example, how come x + 5y becomes x - 5y? The question basically says to let T be the transformation induced by an invertible 2x2 matrix A. In each case, interpret T^-1 geometrically. For this question,

A = 1 5
0 1

http://img332.imageshack.us/img332/7757/ex3re.jpg

Also, just to note that in my initial post, the matrix A for that one was

A = 2 0
0 1

Last edited by a moderator:
I suspect the answer will be clear once you interpret the transformation geometrically.

You could, of course, verify it by turning the crank: check that T T-1 is the identity.

Hurkyl said:
I suspect the answer will be clear once you interpret the transformation geometrically.

You could, of course, verify it by turning the crank: check that T T-1 is the identity.

Dumb question, but how do we get the identity by doing T T-1? Isn't that undefined/does not exist? Unless I'm missing something here.

T and T^-1 are both functions R^2 --> R^2, right? I don't see any obstacle to composing them.

(Or equivalently, multiplying their matrix representations)

Technically, you should prove both T T^-1 and T^-1 T are both the identity.

Alright, thanks.

## 1. What is the inverse of a linear transformation?

The inverse of a linear transformation is a mathematical operation that undoes the effects of the original transformation. It essentially reverses the direction of the transformation, returning the original input values.

## 2. How is the inverse of a linear transformation calculated?

The inverse of a linear transformation can be calculated using a variety of methods, such as the Gauss-Jordan elimination method or the adjoint matrix method. The specific method used depends on the dimension and complexity of the transformation.

## 3. Can all linear transformations have an inverse?

No, not all linear transformations have an inverse. Only transformations that are one-to-one and onto (also known as bijective) have an inverse. This means that each output value has a unique input value and vice versa.

## 4. What is the importance of the inverse of a linear transformation?

The inverse of a linear transformation is important because it allows us to reverse the effects of a transformation and retrieve the original input values. It is also used in various mathematical and scientific fields, such as computer graphics, economics, and physics.

## 5. How is the inverse of a linear transformation used in real-world applications?

The inverse of a linear transformation is used in various real-world applications, such as image and signal processing, data compression, and optimization problems. It is also used in industries like finance and engineering to solve complex problems and make predictions based on given data.

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