Linear Algebra Lecture 9.

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Presentation transcript:

Linear Algebra Lecture 9

Systems of Linear Equations

Linear Transformations

Matrix Equation Vector Equation

Observe

A Transformation or Function or Mapping

A transformation T from Rm is a rule that assigns to each vector x in Rn a vector T(x) in Rm . The set Rn is called the domain of T, and Rm is called the co-domain of T.

The notation indicates that the domain of T is Rn and the co-domain is Rm. For x in Rn , the vector T(x) in Rm is called the image of x (under the action of T). The set of all images T(x) is called the range of T

Example 1

Example 2

Example 2

Example 3

A transformation (or mapping) T is linear if: Definition A transformation (or mapping) T is linear if: T(u + v) = T(u) + T(v) for all u, v in the domain of T; T(cu) = cT(u) for all u and all scalars c.

If T is a linear transformation, then T(0) = 0, and Further If T is a linear transformation, then T(0) = 0, and T(cu +dv) = cT(u) + dT(v) for all vectors u, v in the domain of T and all scalars c, d.

Generally

Examples

Linear Algebra Lecture 9