Linear Transformations problem

In summary, a linear transformation problem involves transforming vectors using a linear function, such as scaling, rotating, and reflecting. This is represented by a matrix, with columns representing transformed basis vectors and rows representing coefficients. A linear transformation preserves properties of linearity, while non-linear transformations do not. They are used in various fields, such as computer graphics and economics, to model and analyze systems and data. Linear transformations have key properties such as preserving vector addition and scalar multiplication, the origin, and lines and planes. They also have a unique inverse and can be composed to create new transformations.
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Homework Statement


Determine if the transformation T: [tex]R^{2}\rightarrow R^{2}[/tex] is linear if T(x, y)= (x+1, 2y)


Homework Equations


1. T(u + v) = T(u) + T(v)
2. T(c*u) = cT(u)
3. T(0) = 0

The Attempt at a Solution


I believe I have to use the above provided equations to determine whether T(x, y)= (x+1, 2y) is linear or not. If I use the third equation above, I get T(0,0) = T(0 + 1, 0) = (1, 0). Therefore the transformation is not linear. Am I right?
 
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Correct.
 

1. What is a linear transformation problem?

A linear transformation problem involves transforming a vector or set of vectors using a linear function. This can include operations such as scaling, rotating, and reflecting the vectors in a given space.

2. How is a linear transformation represented?

A linear transformation can be represented by a matrix. The columns of the matrix represent the transformed basis vectors and the rows represent the coefficients used to transform the original vector(s).

3. What is the difference between a linear transformation and a non-linear transformation?

The key difference between a linear and non-linear transformation is that a linear transformation preserves the properties of linearity, such as parallelism and proportionality. Non-linear transformations do not follow these properties and can have curved or nonlinear outputs.

4. How are linear transformations used in real-world applications?

Linear transformations are used in various fields such as computer graphics, economics, and physics. They can be used to model and analyze systems and processes, and also to create visual representations of data.

5. What are some key properties of linear transformations?

Linear transformations have several important properties, including the preservation of vector addition and scalar multiplication, the preservation of the origin, and the preservation of lines and planes. They also have a unique inverse transformation and can be composed to create new transformations.

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