Question regarding multiple steps in linear mapping

In summary, linear mapping is a mathematical function that maps one vector space to another in a linear way, preserving operations of vector addition and scalar multiplication. Multiple steps in linear mapping can be represented using composition, where matrices are multiplied together. The main difference between linear and non-linear mapping is that linear mapping preserves operations while non-linear mapping does not. The number of steps in linear mapping is determined by the dimension of the vector spaces involved. Linear mapping can also be applied to non-numeric data as long as it can be represented as numerical vectors.
  • #1
blhhblah
6
0
Hi, I'm having some difficulty with this problem. I need to project a point in R2 to the line x2 = x1 (sqrt(3)) and then rotate it 30 degrees clockwise.

I believe the 2x2 matrix to map it is just

sqrt(3) 0
0 1

and to rotate a vector clockwise as opposed to counter clockwise I think is

cosx sinx
-sinx cosx

I could be wrong about these 2 matrices. Is there anyway to merge the two? Do I multiply them? Thanks.
 
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  • #2
Yes, you "merge" two matrices by multiplying them. Be sure to multiply in the right order.
 

1. What is linear mapping?

Linear mapping, also known as linear transformation, is a mathematical function that maps one vector space to another in a linear way. It preserves the operations of vector addition and scalar multiplication, and can be represented by a matrix.

2. How do you represent multiple steps in linear mapping?

To represent multiple steps in linear mapping, we can use composition. This means applying one linear mapping after another, in the order that they are given. This can be represented by multiplying the matrices of each individual linear mapping together.

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

The main difference between linear and non-linear mapping is that linear mapping preserves the operations of vector addition and scalar multiplication, while non-linear mapping does not. In linear mapping, the output is always a scaled version of the input, whereas in non-linear mapping, the output can be a combination of different inputs.

4. How do you determine the number of steps needed in linear mapping?

The number of steps needed in linear mapping is determined by the dimension of the vector spaces involved. The number of steps is equal to the number of rows in the matrix representing the linear transformation. For example, a 2-dimensional vector space would require a 2x2 matrix for linear mapping.

5. Can linear mapping be applied to non-numeric data?

Yes, linear mapping can be applied to non-numeric data as long as the data can be represented in a vector space. This includes data such as text, images, and audio. The key is to find a way to represent the data as numerical vectors that can be transformed by linear mapping.

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