# Linear Transformations matrix help

• DanielJackins
In summary, the conversation discusses two questions that involve finding the image of an arbitrary vector under a linear transformation and finding the matrix representation of the transformation. For the first question, the transformation is given and the image of the arbitrary vector is found to be a 2x2 matrix. For the second question, the cross product of vectors in R3 is defined and used to find the matrix representation of the transformation.
DanielJackins

## Homework Statement

Two questions;

1. Let v1 = [-3, -4] and v2 = [-2, -3]

Let T: R^2 -> R^2 be the linear transformation satisfying T(v1) = [29, -35] and T(v2) = [22, -26]

Find the image of the arbitrary vector [x, y]

T[x,y] = [ _ , _ ]

2. The cross product of two vectors in R^3 is defined by [a1,a2,a3] x [b1,b2,b3] = [a2b3-a3b2, a3b1-a1b3, a1b2-a2b1]

Let v = [2,6,-4]

Find the matrix A of the linear transformation from R^3 to R^3 given by T(x) = v x x.

A = ? (3x3 matrix)

## The Attempt at a Solution

For question 1, I found a T[x,y] but it's a 2x2 matrix [{-197, -8/7},{135,8/7}], but maybe I'm understanding the question wrong?

And for question 2, I really have no idea where to start. Wouldn't I need to be provided with a matrix x if were to find T(x) = v x x? Thanks for any help

You are confusing the representation of the transformation as a matrix with its result as an operator. The domain and range of the operator in 1 is R2. I didn't check your calculations in 1, but if the transformation is represented by the matrix you say it is, you should be able to calculate T([x,y]), which should be in R2.

You got the matrix in part 1 by knowing what T did to a couple of points in R2. Do the same thing for part 2.

## 1. What is a Linear Transformation?

A linear transformation is a mathematical function that maps one vector space onto another, preserving the basic structure of the space. It is represented by a matrix and is used to transform points or objects in one space to points or objects in another space.

## 2. How do you represent a Linear Transformation with a matrix?

A linear transformation can be represented by a matrix by defining a standard basis for both the input and output vector spaces. The columns of the transformation matrix are the images of the basis vectors of the input space under the transformation.

## 3. What is the purpose of a Linear Transformation?

A linear transformation is used to change the orientation, shape, or size of an object or space. It is commonly used in fields such as computer graphics, physics, and engineering to transform and manipulate data.

## 4. What properties do Linear Transformations have?

Linear Transformations have a few key properties, including preserving lines and planes, preserving the origin, and preserving vector addition and scalar multiplication. They also have the property of being commutative, meaning that the order of transformations does not affect the end result.

## 5. How do you apply a Linear Transformation to a vector?

To apply a linear transformation to a vector, you simply multiply the transformation matrix by the vector. The resulting vector will be the transformed version of the original vector. This process can also be extended to apply the transformation to multiple vectors at once by using a matrix of column vectors.

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