Linear Algebra

Basis and Dimension

In the previous chapter, we saw how the concept of minimal spanning sets leads naturally into linear independence and degrees of freedom. In this chapter, we’ll take a deeper look into these minimal spanning sets.

Basis and Dimension

First, some terminology: given a vector space \(V\), any minimal spanning set of \(V\) is called a basis. Let’s take a look at the simple vector space \(\mathbb{R}^3\), i.e. triples of real numbers. The “standard” basis of this vector space is the simple \[\left\{\begin{pmatrix}1\\0\\0\end{pmatrix}, \begin{pmatrix}0\\1\\0\end{pmatrix}, \begin{pmatrix}0\\0\\1\end{pmatrix}\right\}.\] However, there are also other possible bases. Suppose we have the basis \[\left\{\begin{pmatrix}1\\0\\0\end{pmatrix}, \begin{pmatrix}0\\1\\0\end{pmatrix}, v\right\}.\] Which of the following could be \(v?\)

                 

Basis and Dimension

The last problem demonstrates a general method for checking if a set is in fact a basis. First, we find a “standard” basis, such as \[\left\{\begin{pmatrix}1\\0\\0\\\vdots\\0\end{pmatrix}, \begin{pmatrix}0\\1\\0\\\vdots\\0\end{pmatrix}, \ldots, \begin{pmatrix}0\\0\\0\\\vdots\\1\end{pmatrix}\right\}\] for \(\mathbb{R}^n\), and then we try to reduce our set to this standard basis by writing all the standard basis vectors as linear combinations of the vectors in our set. For instance, in the previous problem we just needed to write \(\begin{pmatrix}0\\0\\1\end{pmatrix}\) (one of the standard basis vectors) as a linear combination of the vectors in the sets we were checking.

Basis and Dimension

Now let’s move to a related concept: the dimension of a vector space, which is defined as the size of the basis. Technically, this requires two details we’ll skate over for the time being: that every vector space has a basis, and that all finite bases of a vector space have the same size.

For instance, we’ve already seen that the dimension of \(\mathbb{R}^n\) is \(n\). Let’s look at some other vector spaces now. To start, consider the vector space of \(3 \times 3\) matrices. What is its dimension?

Basis and Dimension

Now let’s look at some more interesting vector spaces. A symmetric matrix is a square matrix that is symmetric about its main (top left-bottom right) diagonal, such as \[\begin{pmatrix}1&2&3\\2&1&3\\3&3&1\end{pmatrix}.\] Consider the \(3 \times 3\) symmetric matrices. They form a subspace of the \(3 \times 3\) matrices (why?). What is the dimension of the vector space they form?

Basis and Dimension

Now let’s switch tack to a different vector space: polynomials with degree at most 3, such as \(x^ 3 + x\). These form a vector space (why?). What is the dimension of this space?

Basis and Dimension

We’ve now seen a couple examples of how the dimension of a subspace is less than the dimension of the original space, which makes dimension a useful tool for thinking about the “size” of a vector space.

In fact, more is true: given a basis \(B’\) for a subspace \(V’\) of \(V\), we can extend the basis by adding more vectors to form a basis \(B\) of \(V\).

This becomes important when analyzing bases of subspaces, because we can pick a basis for the subspace, and then pick a basis for the larger space that is guaranteed to be a superset of the original basis.

Basis and Dimension

Let’s return to symmetric matrices from before. As a reminder, a symmetric matrix is a square matrix that is symmetric about its main (top left-bottom right) diagonal, such as \[\begin{pmatrix}1&2&3\\2&1&3\\3&3&1\end{pmatrix}.\] Let’s also look at what we’ll call negative symmetric matrices, which are similar except that the non-diagonal entries are the negative of their symmetric counterpart, such as \[\begin{pmatrix}1&2&3\\-2&1&3\\-3&-3&1\end{pmatrix}.\] Suppose the symmetric matrices form the vector space \(V\), and the negative symmetric matrices form the vector space \(W\). As we’ve seen, both \(V\) and \(W\) have dimension 6. What are the dimensions of the vector spaces \(V \oplus W\) and \(V \cap W?\)


Recall \(V \oplus W\) consists of all vectors \(v + w\) where \(v \in V\) and \(w \in W\), while \(V \cap W\) consists of all vectors \(v\) for which \(v \in V\) and \(v \in W\).

                 

Basis and Dimension

In this chapter, we continued our explorations into bases, and also explored the related concept of dimension. As we saw, dimension is a very useful tool for analyzing both subspaces and “indirectly defined” vector spaces, such as the important relation \[\text{dim}(V \oplus W) + \text{dim}(V \cap W) = \text{dim}(V) + \text{dim}(W).\] In the next chapter, we’ll move past our exploration of basis and dimension on to a different important concept: the dot product.

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