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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?
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.
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.
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.
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.
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.