Linear AlgebraLinear Transformations
Functions describe relationships between sets and thereby add dynamism and expressive power to set theory. Likewise, linear transformations describe linearity-respecting relationships between vector spaces. They are useful for understanding a variety of vector space phenomena, and their study gives rise to generalization of the notion of linear dependence which is very useful in numerical applications of linear algebra (including describing the structure of real-world datasets).
A linear transformation is a function from one vector space to another which satisfies . Geometrically, these are "flat maps": a function is linear if and only if it maps equally spaced lines to equally spaced lines or points.
In reflection along the line defined by , is linear because
Many fundamental geometric transformations are linear. The figure below illustrates several linear transformations (as well as one nonlinear one, for comparison) from the plane to the plane. The leftmost column shows a square grid of points, and the rightmost column shows the images of those points. The other columns show each point somewhere along the path from its original location in the domain to its final location in the codomain, to help you get a sense of which points go where.
This 3Blue1Brown video provides some helpful animated illustrations of linear transformations:
The rank of a linear transformation from one vector space to another is the dimension of its range.
If , then the rank of
Find the rank of the linear transformation
Solution. The range of
What are the ranks of the five transformations illustrated above?
- The rank of the rotation is
- The rank of the reflection is
- The rank of the scaling transformation is
- The rank of the shearing transformation is
- The rank of the projection is
The null space of a linear transformation is the set of vectors which are mapped to the zero vector by the linear transformation.
Note that the range of a transformation is a subset of its
Because linear transformations respect the linear structure of a vector space, to check that two transformations from a given vector space to another are equal, it suffices to check that they map all of the vectors in a given basis of the domain to the same vectors in the codomain:
Exercise (basis equality theorem) Suppose that
What is the dimension of the null space of the linear transformation
The dimension of the null space is
Solution. To find the dimension of the nullspace, let us first describe it explicitly.
We call the dimension of the null space of a linear transformation the nullity of the transformation. In the previous exercise, the rank and the nullity of
Theorem (Rank-nullity theorem) If
Proof. If we
is a basis for the
These vectors are linearly independent because
which in turn implies that
To see that
by linearity of
Since the list
Suppose you're designing an app that recommends cars. For every person in your database, you have collected twenty values: age, height, gender, income, credit score, etc. In your warehouse are ten types of cars. You envision your recommendation system as a linear transformation
After some time, you find that storing all twenty variables takes up too much space in your database. Instead, you decide to take those twenty variables and apply a linear aggregate score function
Solution. The image of the transformation