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What is the linear transformation from R³ to R²?

1) A transformation that maps a three-dimensional vector to a two-dimensional vector
2) A transformation that maps a two-dimensional vector to a three-dimensional vector
3) A transformation that maps a three-dimensional vector to another three-dimensional vector
4) A transformation that maps a two-dimensional vector to another two-dimensional vector

1 Answer

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Final answer:

A linear transformation from R³ to R² is a function that maps three-dimensional vectors to two-dimensional vectors.

Step-by-step explanation:

The correct answer is 1) A transformation that maps a three-dimensional vector to a two-dimensional vector. A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. When we refer to a transformation from (three-dimensional real number space) to (two-dimensional real number space), we are describing a function that takes a vector with three components and returns a vector with two components.

For example, a matrix A that is 2x3 can represent such a transformation. If v is a vector in , the product Av will be a vector in . This is achieved by multiplying each element of the vector in by the corresponding elements in matrix A and summing up the results to produce the entries of the vector in .

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