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.


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.


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.


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.


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.

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