Linear transformation T: R3 -> R2

In summary, a linear transformation T: R3 -> R2 is a mathematical function represented by a 2x3 matrix that maps a vector in three-dimensional space to a vector in two-dimensional space, while preserving the operations of addition and scalar multiplication. It is commonly used in various real-world applications such as engineering, physics, computer graphics, data analysis, and machine learning. To determine if a given transformation is linear, one can test it on two vectors and see if the result is equal to the sum or scalar multiple of the individual transformations of the vectors.
  • #1
GrafZeppelim
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TL;DR Summary
Linear transformation T: R3 --> R2

Homework Statement

Find the linear transformation [/B]
T: R3 --> R2 such that:

𝑇(1,0,−1) = (2,3)
𝑇(2,1,3) = (−1,0)

Find:

𝑇(8,3,7)
Does any help please?
 
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  • #2
Express (8,3,7) as a linear combination of the two other vectors in the problem statement and apply linearity of the transformation.
 

1. What is a linear transformation?

A linear transformation is a mathematical function that maps a vector from one vector space to another in a way that preserves the structure of the original space. In simpler terms, it is a function that takes in a vector and outputs another vector, while maintaining the same direction and scaling.

2. How is a linear transformation represented?

A linear transformation can be represented by a matrix. In the case of a transformation from R3 to R2, the matrix will have 3 rows and 2 columns. The elements of the matrix represent how the transformation affects each component of the input vector.

3. How do you perform a linear transformation on a vector?

To perform a linear transformation on a vector, you multiply the vector by the transformation matrix. The resulting vector will be the transformed vector in the new vector space. For a transformation from R3 to R2, the vector will have 2 components instead of 3.

4. Can a linear transformation change the dimension of a vector space?

Yes, a linear transformation can change the dimension of a vector space. In the case of a transformation from R3 to R2, the dimension of the output vector space will be 2, while the dimension of the input vector space is 3.

5. What is the importance of linear transformations in science?

Linear transformations are important in science because they allow us to model and understand real-world phenomena in a mathematical way. They are used in various fields such as physics, engineering, and computer science to solve problems and make predictions.

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