Give an example of a linear transformation

In summary, a linear transformation is a mathematical function that preserves the structure of a vector space, such as distances and angles between vectors, and can be represented as a combination of operations like scaling, rotation, reflection, and shearing. Some common examples include scaling, rotation, reflection, shearing, projection, and translation, which are used in computer graphics, image processing, and engineering applications. A real-life example of a linear transformation is the conversion of a 2D object into a 3D object. Linear transformations differ from non-linear transformations in that they preserve the basic structure of a vector space and can be represented as a linear combination of operations. Some practical applications of linear transformations include modeling real-world systems, image processing, computer graphics
  • #1
62
0

Homework Statement


Give an example of a linear transformation whose kernel is the line spanned by:

-1
1
2

in lR³

Homework Equations





The Attempt at a Solution



Would:

1..(-1)...0
0...0...0
0..(-2)...1

be a solution?
 
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  • #2
To prove it, prove your vector is in the kernel and that the dimension of the col. space is 3 - nullity = 2.
 
  • #3
It quite obviously isn't since Av is not 0. Are you clear on the definition of "kernel"?

Actually your matrix has only a single entry wrong so it may be just a typo.
 

What is a linear transformation?

A linear transformation is a mathematical function that maps a vector space onto itself in a way that preserves the basic structure of the space, such as the distances and angles between vectors. It can be represented as a matrix multiplication or a combination of scaling, rotation, reflection, and shearing operations.

What are some common examples of linear transformations?

Some common examples of linear transformations include scaling, rotation, reflection, shearing, projection, and translation. These operations are commonly used in computer graphics, image processing, and engineering applications.

Can you give an example of a linear transformation in real life?

One example of a linear transformation in real life is the transformation of a 2D object, such as a map or a painting, into a 3D object, such as a globe or a sculpture. This transformation involves scaling, rotation, and translation operations that preserve the basic structure of the original object.

How is a linear transformation different from a non-linear transformation?

A linear transformation is a type of transformation that preserves the basic structure of a vector space, while a non-linear transformation does not. In other words, a linear transformation can be represented as a linear combination of basic operations, while a non-linear transformation cannot.

What are some practical applications of linear transformations?

Linear transformations have many practical applications in fields such as computer science, engineering, physics, and economics. They are used to model and analyze real-world systems, perform image processing and computer graphics operations, and solve optimization problems.

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