In the previous section, we discussed the concept of vector spaces, which are collections of objects that are closed under addition and scalar multiplication. As we saw, matrices, functions, and magic squares all form vector spaces, as do more advanced concepts like solutions to differential equations.

In this quiz, we’ll look at breaking down vector spaces into smaller **subspaces**, which are vector spaces that are contained entirely within the original vector space. Intuitively, a subspace is a “special case” (or more formally a subset) of its containing vector space. In fact, you may have already noticed an example from the previous quiz: the vector space of magic squares is a subspace of the \(3 \times 3\) matrices.

Formally, a vector space \(V’\) is a **subspace** of a vector space \(V\) if

- \(V’\) is a vector space;
- every element of \(V’\) is also an element of \(V.\)

Because \(V’\) is a subset of \(V\), we can skip checking most of the vector space axioms from before. All we need is

- 0 is in \(V’;\)
- for all \(u, w\ \in V’\), \(u + w \in V’;\)
- for all \(v \in V’\), \(cv \in V’\) for any constant \(c.\)

Note that ordered pairs of real numbers form a vector space \(V\). Which of the following is a subspace of \(V?\)

By now, we have a good bit of practice verifying (or disproving) that some set is a vector space. Now, instead of starting with a set and checking if it’s a vector space, let’s take a slightly different approach and try to “build” a vector space directly.

Keeping in mind the closure axioms (addition and scalar multiplication), which of the following is a vector space that contains both the vectors \((4,2)\) and \((2,4)?\)

In the previous problem, we saw how to “build” a vector space containing certain vectors. Specifically, if we have a set of vectors \(\{v_1, \ldots, v_n\}\), then the set of \[c_1v_1 + c_2v_2 + \ldots + c_nv_n\] for constants \(c_1, \ldots, c_n\) is the “smallest” (we’ll define rigorously what “smallest” means in the context of vector spaces later) vector space containing these vectors. Make sure you understand why this is a vector space!

This concept is so important that we give it a special name: this vector space is called the **span** of the set of vectors. Another way of thinking about the span is that any element of a vector space can be written as a linear combination of the elements in the space’s spanning set.

The reason span is so important is because it allows us to easily identify subspaces. In particular, if a span of a vector space is a subset of a span of another vector space, then the first vector space is a subspace of the second. For instance, the vector space spanned by \((1,1)\) is a subspace of the vector space spanned by \((1,1)\) and \((1,2)\).

Consider the following three vector spaces:

A)The vector space spanned by \((1,2)\)

B)The vector space spanned by \((1,2)\) and \((2,3)\)

C)The vector space spanned by \((1,2)\), \((2,4)\), and \((3,6)\)

Which of the following is true?

Let’s now look at a couple other operations on subspaces. The **intersection** of two vector spaces \(V\) and \(W\), written \(V \cap W\), is exactly what you’d expect: the set of vectors that are in both \(V\) and \(W\). The **sum** of two vector spaces \(V\) and \(W\), written \(V \oplus W\), is slightly trickier: the set of vectors \(v + w\), where \(v \in V\) and \(w \in W\). Finally, the **union** of two vector spaces, written \(V \cup W\) is the set of vectors in either \(V\) or \(W\).

Suppose that \(V_1\) and \(V_2\) are both subspaces of \(V\). Which of the following is **not** necessarily a subspace of \(V?\)

In this quiz we looked at **subspaces** of vector spaces, and how to characterize them using the **span** of sets of vectors. As we saw, sometimes the span has “redundant” vectors that make things quite confusing! In the next quiz, we’ll look more at *minimal* spanning sets, which will also allow us to talk about things like the “size” of vector spaces… and allow us to finally formalize the “degrees of freedom” concept we noticed long ago.

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