3.1 Linear Transformaitons

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9 Terms

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Codomain

A set containing all possible outputs for a function. (Note that this contains the range, which is equal to the set of possible outputs for a function.)

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Domain

The set of possible inputs for a function

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Image

The output of a function from a particular input

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Linear transformation

A function T : Rm Rn is a linear transformation if for all vectors u and v in Rm and all scalars r we have;

T(u + v) = T(u) + T(v)

T(ru) = rT(u)

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Matrix dimensions

The number of rows and columns for a matrix. Generally displayed as n × m, where n is the number of rows and m the number of columns

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One-to-one

A linear transformation T : Rm → Rn is one-to-one if for every vector w in Rn there exists at most one vector u in Rm such that T(u) = w. Alternate definition: A linear transformation T is one-to-one if T(u) = T(v) implies that u = v.

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Onto

A linear transformation T : Rm → Rn is onto if for every vector w in Rn there exists at least one vector u in Rm such that T(u) = w

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Range

The set of outputs for a function

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Square matrix

matrix with the same number of rows and columns