# Finding the Transformation Matrix if it's linear

• silvermane
In summary, the transformations a and b are linear, with transformation matrices being a row operation and an identity map, respectively. The transformation c is also linear and represents a projection. However, the transformation d is not linear as all values cannot map to the same point. A transformation matrix cannot be found for d as it does not satisfy the requirements for linearity.
silvermane
Gold Member

## Homework Statement

Which of the transformations are linear? If they are, then find the transformation matrix.
the input is v = (v1,v2)

a. t(v) = (v2,v1)
b. t(v) = (v1,v2)
c. t(v) = (0,v1)
d. t(v) = (0,1)

## The Attempt at a Solution

a. it is linear
b. it is linear
c. I think it is linear because we're going from R2 to R1 and not going to the origin.
d. I don't think this one is linear, because all values can't map to the same point.

Now, to find the transformation matrix A, I would like a helpful tip/hint/procedure for how to find the matrix A. I am having some difficulty understanding this concept, as it's not in the book and this part was added by the teacher.

(I'm off to bed, but will be back in the morning, hehe)

The first one is simply a row operation of switching rows.

The second one is an identity map.

The third is a projection.

How about the last one ? It is not linear. Can you find a transformation matrix ? Remember every matrix transformation is linear.

All these a standard transformations with their matrices in most books or google. You should be able to come up with them one your known .

You can find matrices such that when you multiply your column vector by them you get the desired result. It shouldn't be too difficult.

Going to bed too.:-)

Last edited:
silvermane said:
d. t(v) = (0,1)

d. I don't think this one is linear, because all values can't map to the same point.

The result is correct, but your reasoning is not. t(v) = (0,0) is (trivially) linear, and it maps everything to the same point.

*Awaken*

The man is right; pershaps, you should show that the maps fails one of the test of linearity.

## 1. What is a transformation matrix?

A transformation matrix is a mathematical representation of a linear transformation, which is a function that maps points from one coordinate system to another. It is typically represented as a square matrix and is used to describe geometric transformations such as rotations, translations, and reflections.

## 2. How do you know if a transformation is linear?

A transformation is considered linear if it preserves the properties of linearity, such as scaling, addition, and multiplication by a scalar. This means that the output of the transformation can be calculated by applying the transformation to each individual component of the input vector and then combining the results.

## 3. Can a transformation matrix be non-square?

No, a transformation matrix must be square in order to represent a linear transformation. This is because it must be able to map points from one coordinate system to another, and this requires the same number of dimensions in both systems.

## 4. How do you find the transformation matrix for a specific transformation?

To find the transformation matrix, you need to know the effects of the transformation on a set of basis vectors. These are typically the unit vectors along the x, y, and z axes. By applying the transformation to these basis vectors and arranging the resulting vectors as columns in a matrix, you can obtain the transformation matrix.

## 5. What happens if the transformation matrix is not invertible?

If the transformation matrix is not invertible, it means that the transformation does not have a unique inverse. This can happen when the transformation maps multiple points to the same location, making it impossible to retrieve the original points. In this case, the transformation is not reversible and is known as a non-invertible or singular transformation.

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