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Linear Transformations

A linear transformation is a special function, a function that maps an object in one vector space to an object in another vector space (Note: the two vector spaces can be the same vector space.)

Let V and W be vector spaces with ordered bases β and γ respectively:

                  β= x1 xN    γ= y1 yM

Let T be a linear transformation from V into W:

                  T:VW

The transformation T takes any vector x in V and maps it to a vector Tx in W:

                  xTx

For example, the vectors in the basis β are mapped as follows:

                  xjTxj

                  Txj=i=1Maijyi 1jN

The above expression maps the basis vector xj of V to the vector Txjin W. The vector Txj is computed as a linear combination of the M basis vectors in the basis γ of W; each basis vector contributes to the combination by an amount determined by the (unique) real number aij. Therefore, there are M such numbers for each Txj that corresponds to xj.

The coefficients aij in the linear combinations above form a matrix A which we call the transformation matrix that represents T. Each column of this matrix is a coordinate vector that corresponds to a basis vector in the basis β of the vector space V. More specifically, the jth column of this matrix represents the coordinate vector Txj γ of Txj corresponding to the basis vector xj:

                  A= Tx1 γ TxN γ

We also use the notation T βγ to denote this matrix in order to stress the fact that T is a linear transformation with respect to the ordered bases β and γ.

                  A= T βγ and Aij=aij where 1iM and 1jN

The following are important facts worth remembering about the transformation matrix A.

Each column of A represents a coordinate vector corresponding to a basis vector in β of V. Namely, the jth column of A represents the coordinate vector Txj γ of Txj corresponding to the basis vector xj in the basis β of V.

The transformation matrix A has N columns, because its jth column represents the coordinate vector of Txj corresponding to the basis vector xj and there are N such basis vectors.

The transformation matrix A has M rows, because its jth column represents the coordinate vector Txj γ of Txj corresponding to the basis vector xj and the coordinate vector has M elements.

Examples

Click here to see examples of linear transformations.

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