Lecture Notes

2. Subspaces

Part of the Series on Linear Algebra.

By Akshay Agrawal. Last updated Sept. 3, 2017.

Previous entry: Vector Spaces; Next entry: Basis and Dimension

In the previous entry, we studied vector spaces, sets in which addition and scalar multiplication satisfy some intuitive properties. In this section, we study a special type of vector space: the subspace. These notes are adapted from Sheldon Axler’s excellent introductory text Linear Algebra Done Right.

2.1 Definition

Definition 2.1 A subspace of a vector space $V$ is a subset of $V$ that is itself a vector space. To check whether a subset $U$ of a vector space is indeed a subspace, it is sufficient and necessary to check that $0 \in U$, $U$ is closed under addition, and $U$ is closed under scalar multiplication, i.e., $U$ is a subspace of $V$ if and only if these three conditions are satisfied.

If a set $U$ is a subspace (or, for that matter, a vector space), you immediately know that it posseses a nice additive and multiplicative structure. For example, it is a fact that the set of differentiable real-valued functions on the reals is a subspace of $\Reals^\Reals$ — this tells us, for example, that the sum of two real-valued differentiable functions is itself differentiable.

2.2 Sums of subspaces

Definition 2.2 The sum of two subsets $U_1$ and $U_2$ of a vector space $V$ is the set that contains all possible sums of their elements: $U_1 + U_2 = \{u_1 + u_2 : u_1 \in U_1, u_2 \in U_2\}$ (the sum of more than two sets is defined analogously). The summation of sets in this way is sometimes called the Minkowski sum.

Definition 2.3 A sum of subspaces $U_1$ through $U_m$ of a vector space is a direct sum if each element in $U_1 + \ldots + U_m$ is uniquely determined by the sum of some $u_1, u_2, \ldots u_m$, with each $u_i \in U_i$, ie, if there is only one way to write each $u \in U$ as a sum of elements from the $U_i$. If the sum $U_1 + U_2 + \ldots + U_m$ is in fact a direct sum, it is common to emphasize this fact by substituting the “$\oplus$” symbol for the “$+$” symbol, like so: $U_1 \oplus U_2 \oplus \ldots \oplus U_m$.

It is a useful and easy to verify fact that the sum of two subspaces is a direct sum if and only if their intersection is the singleton set $\{0\}$.

2.3 Exercises

1. Verify the claim made at the end of §2.2: $U + W \equiv U \oplus W$ if and only if $U \cap W = \{0\}$, where $U$ and $V$ are subspaces of the same vector space.
2. Prove that $U_1 + \ldots + U_m$, where the $U_i$ are subspaces of the same vector space, is a direct sum if and only if the only way to write $0$ as a sum of the $U_i$ is by taking each term in the sum to be $0$.

2.4 References

Linear Algebra Done Right, by Sheldon Axler.