Linear Transformation Problem

In summary, the conversation is about a year 10 student seeking help with two problems in a book on Transformation. They have a sudden exam in three days and are looking for assistance or appropriate books to read. The first problem is about determining the matrix of a linear transformation based on the images of two points, while the second problem involves finding the matrix and range of a specific linear transformation. The conversation ends with the student thanking the reader and providing additional information on the nature of linear transformations.
  • #1
LovePhys
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Hello everyone, I am a year 10 student. I am working on Transformation and there are 2 problems in the book that I actually have no idea to solve. Unfortunately, I am going to have a sudden exam in the next three days.

Hope that somebody can help me, thank you a lot! Or If you have any books which is appropriate with high school students, I am glad to read it!

1/ The images of 2 points are given for a linear transformation. Investigate whether this is sufficient information to determine the matrix of the transformation.
2/ Find the matrix of the linear transformation such that (1,0) -> (1,1) and (0,1) -> (2,2). What is the range of this transformation? (I found the matrix of the linear transformation).

Thank you,
Huyen Nguyen
 
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  • #2
1/ start by trying to write down an arbitrary linear transformation based on the known image of 2 known points

2/ note tht every point will bee mapped onto a line - can you find the equation of the line
 

What is a linear transformation?

A linear transformation is a mathematical operation that maps one vector space into another vector space in a linear way. This means that the output of the transformation is a linear combination of the input vectors.

What is the purpose of solving linear transformation problems?

The purpose of solving linear transformation problems is to understand and analyze the behavior of vectors and matrices under linear transformations. This can be useful in various fields such as physics, engineering, and computer science.

What are some common examples of linear transformations?

Some common examples of linear transformations include rotation, scaling, shearing, and reflection. These transformations can be represented by matrices and can be applied to vectors to produce a new vector.

What are the key properties of linear transformations?

The key properties of linear transformations include preserving the zero vector (T(0) = 0), preserving addition (T(u + v) = T(u) + T(v)), and preserving scalar multiplication (T(ku) = kT(u)). These properties allow for the analysis and manipulation of linear transformations in a systematic way.

How do you solve a linear transformation problem?

To solve a linear transformation problem, you typically need to determine the transformation matrix and then apply it to the given vector. This will result in a new vector, which can then be analyzed to understand the effect of the linear transformation. Depending on the problem, you may also need to apply other operations such as matrix multiplication or inverse operations.

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