Basis and dimension

A basis is a set of linearly independent vectors (for instance $\vec{v_1}, \ldots \vec{v}_n$) that span a vector space or subspace. That means that any vector $\vec{x}$ belonging to that space can be expressed as a linear combination of the basis for a unique set of constants $k_1, \ldots k_n$, such as: $$\vec{x} = k_1\vec{v}_1 + \ldots + k_n\vec{v}_n$$ The dimension of the vector space corresponds to the number of vectors required to form a basis (the basis is not unique). In this example, $n$.

Intro

Genes are the fundamental units of living organisms that explain the functionality of any species, which is why they are of such importance for scientists trying to explain certain biological traits.

Humans have over twenty thousand different cells, each contributing to our impressive abilities. When abnormalities emerge, such as chronic diseases, the explanation and potential cure is often to be found in the divergent behavior of individual genes and their interactions.

A human patient can be represented by a vector, where each component corresponds to a separate gene. With a method called principal component analysis, we can rewrite these vectors so that each component represent a combination of genes that tend to work together.

This procedure is an example of a change of basis, and can help scientists determine what genes are linked with a certain disease.

Concept

Let's think about what it means for a vector to have the components and . We will right away picture the Cartesian coordinate system with a horizontal x-axis and a vertical y-axis, and draw the vector from the origin to the point .

This way of interpreting a vector is intuitive and convenient, but it is not our only option. We can in fact define each component as any number of steps in any particular direction, by constructing a different basis.

A basis is namely a list of vectors that define the direction and step size of the components of the vectors in that basis. The number of basis vectors hence equals the number of components for vectors in such basis, defining their dimension.

Math

The standard unit vectors and constitute the standard basis of .

Therefore, the general form of a vector in two dimensions represents a linear combination of these particular vectors:

Following this same format, we can write the same vector in another basis , having the basis vectors and :

From these equations, we can derive the relationship necessary for the change of basis by equating the two expressions for :

Basis and coordinates

Standard basis

Let's take a proper step back and go through the coordinates of a vector in as if it were the first time, like when we learned to read a diagram in a coordinate system in elementary school. We have previously mentioned the standard unit vectors and :

It is clear that we can describe as a linear combination of and as follows:

where the coordinates, and , constitute scalars in the linear combination. Any vector can be expressed in as a linear combination of and , and therefore we say that and form a basis for . This is because they are linearly independent and together span the whole space. This can be written algebraically as:

This can be extended analogously to all vectors and subspaces as defined below:

A set of linearly independent vectors in the subspace in are said to constitute a basis for if they span , that is, if:

It is clear that , form a basis for , and are called the standard basis. In addition, these vectors are orthogonal to eachother with length 1, and are therefore said to constitute an orthonormal basis, or ON-basis. If they were only orthogonal, they would have fulfilled the condition of an orthogonal basis.
Properties of each non-null subspace in are:

• There is a basis for that has at most vectors.

• Each basis of has the same number of basis vectors.

Coordinates regarding other bases

The reason for the rigorous review of the standard basis and the definition of basis is to introduce the beginner to working with other bases. Remember what applied to the coordinates of an arbitrary vector ! The coordinates are the scalars in the linear combination of expressed in the basis vectors. Let:

be the basis of a subspace of . Then we have that any arbitrary vector expressed in has the coordinates:

such that:

The above leads to a linear system of equations where the number of equations is (because the basis vectors belong to ), and the number of unknowns is (= number of coordinates). The solution set will be a unique solution given that (although ).
Note that we always have the same number of coordinates as the number of basis vectors spanning the subspace.

Example 1

Let:

be expressed in the standard basis . Also let:

be a basis for . We are now looking for expressed in base . It is highly recommended that the beginner always start their solutions for similar problems in the following way:

We are looking for:

such that:

Starting with these two lines is a proven concept among thousands of students, which makes the section on coordinates with respect to other bases understandable with very good results. From the second line we can derive a linear system of equations with a unique solution:

From the second line, we can convert to the matrix form and augmented matrix and solve the system:

From the row-reduced augmented matrix, we can now read out the solution and the answer for the expression of in the base , which is:

From the image below, we can see how the two vectors and span the parallelogram whose diagonal is , completely according to our start to the solution above and according to the general definition of coordinates.

Example 2

Let:

be a vector in the subspace with parametric form:

We see that spans a plane because it has two vectors and two parameters and . Therefore:

constitutes a basis for the plane / subspace . We are now searching for the coordinates of expressed in base . Again, we start in the following way to solve the problem:

Let:

such that:

We set up the augmented matrix and solve the system:

and we get:

We double-check the result:

Note that we have the same number of coordinates as basis vectors. This means that the vector is expressed by two coordinates, even though exists in . More often than not, unfortunately, this tends to confuse the beginner. Therefore, we draw the following picture to mitigate discomfort.

On the plane we have the vectors , and . We have also included a vector that is not on the plane. All four vectors are found in and therefore have three coordinates. However, all vectors on the plane are expressed with two coordinates when expressed in a basis for the plane. Because the plane is spanned by two vectors, the plane's dimension is two. Therefore, the number of basis vectors is two, and consequently the number of coordinates is two.

According to the picture, we also have the individual Mr. 2D who lives on the plane . He can perceive all the vectors on the plane, including , and . He sees these from his, somewhat limited, perspective. In his world, there are only the concepts width and height, that is, two dimensions. He therefore sees the coordinates of as two and not three. Unfortunately, Mr. 2D cannot perceive the vector , which requires the depth perspective. To reinforce the example with subspaces and local coordinates, we introduce the line that intersects the plane at point . Mr. 2D cannot perceive the line in its entirety, but does have the capcity to perceive the point of intersection , which is also expressed by two coordinates in the subspace in which Mr. 2D lives.

This example is a classic in modern physics to explain the concept of higher dimensions. How do we approach the concepts of space, time and additional dimensional space? Well, in those contexts, it is we who are Mr. 2D, and the possibilities are endless.

Dimension

Dimension and bases are very closely linked and can be linked to our knowledge of the parametric form for the line and the plane. Remember that:

• the parametric form of the line has a parameter, has a direction vector and extends over a dimension.

• the plane's parametric form has two parameters, has two direction vectors and extends over two dimensions.

If a line and a plane intersect, they form subspaces in , and if we replace the word direction vectors with basis vectors, we can connect the comparison of these geometric objects with what we just learned about bases and subspaces. We clearly see the relationship with the number of basis vectors to a subspace and its dimension, which leads us to the following properties for each non-null subspace of :

• The dimension of the subspace is the same as the number of basis vectors.

• A set of more vectors than the dimension of is linearly dependent, and thus does not form a basis of .

• If a set of vectors spans , but is not a basis for , then a basis can be created by removing the appropriate vectors from .

• If a set of vectors is linearly independent but does not span , then a basis can be created by adding the appropriate vectors from to .