Linear Transformations

A linear transformation is a mapping from one vector space to another. It must fulfill the following conditions:

Commutativity: F(x + y) = F(x) + F(y); and

Distributivity: F(λx)= λF(x), where λ is a scalar

The clearest example of a linear transformation is matrix multiplication—L(x)=Ax. Multiplying by a matrix affects a given vector in several ways—rotation, dilation—which includes expansion and contraction, and reflection or inversion. Some matrices have a known and cataloged effect on the vectors they multiply. Most matrices will give a combination of these effects.

Matrix multiplication produces linear transformations from one dimension to another, or within the same dimension. For example, multiplication by a square matrix of dimension n returns a transformation within dimension n. This is because only a vector of the same dimension can be multiplied by such a matrix. An m x n matrix will induce a transformation from Rn to Rm.

Dilations

The most well known transformation is given by the multiplication of a vector with the identity matrix. This returns the original vector unchanged. If In is scaled by α, a vector multiplied by α In will be scaled by α. For α >1, this transformation is an expansion, or enlargement. For α<1, the transformation is a contraction.

Rotations If multiplied by a matrix [cosθ -sinθ, sinθ cosθ] a vector will be rotated counterclockwise by angle θ. As a case in point, the identity matrix represents a rotation by angle 0.

Reflection or Inversion

Multiplication by the matrix [1 0, 0 -1] gives reflection over the x axis in R2. It is possible to multiply by matrices to reflect over axes and lines.

Isometry

If a linear transformation preserves the distances between points in the figure being transformed, it is called an isometry.

How to determine the matrix that causes a linear transformation:

We know that the vectors of the matrix A will be the same as the transformations it gives to the vectors that make up the standard basis for the space from which it is transformed.

A special case of linear transformations is given by eigenvectors and eigenvalues. In a linear transformation over a constant vector space, there will be certain vectors that when multiplied by the matrix A, give a multiple of themselves. The multiple is denoted by the eigenvalue. These special vectors are the eigenvectors of the matrix.

Properties:

Taking derivatives and taking the inner product are functions that give linear transformations.

A line will always be transformed to another line.

A transformation of 0 will give 0.

Isomorphisms: The word derives from the Greek roots iso-, which means “equal” or “same” and morphos which means form, shape or structure. Isomorphic can refer to vector spaces—isomorphic vector spaces contain the same algebraic properties.

Applications: One important application of linear transformations is cryptology.


References:

Breitenbach, Jerome R. "A Mathematics Companion for Science and Engineering Students." New York: Oxford University Press, 2008.

Kolman, Bernard and Hill, David. "Elementary Linear Algebra with Applications and Labs: Custom Edition for Purdue University." Boston: Pearson Learning Solutions, 2004.

Rowland, Todd and Weisstein, Eric W. "Linear Transformation." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LinearTransformation.html

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