Vector spaces · Definition

Vector subspace

linear algebravector spaces

Before you start you need to know

Large vector spaces contain pieces that behave like spaces on their own; recognizing them lets us describe solution sets and work with smaller parts.

Definition

A vector subspace \(W\) of a vector space \(V\) is a nonempty subset that is itself a vector space under the same operations.

\[0\in W,\quad u,v\in W\Rightarrow u+v\in W,\quad c\in\mathbb{R},\ v\in W\Rightarrow cv\in W\]

To decide whether a subset \(W\) of \(V\) is a subspace you do not need to check all eight axioms, since they are inherited from \(V\). Three checks suffice: \(W\) is nonempty (in practice, it contains the zero vector), the sum of two elements of \(W\) stays in \(W\), and any multiple of an element of \(W\) stays in \(W\). The last two are summed up by saying that \(W\) is closed under the operations.

A first filter is the zero vector: if \(0\notin W\) the set is not a subspace, because \(0=0\cdot v\) would have to be in it. So the line \(y=2x\) passes through the origin and is a subspace, while the line \(x+y=1\) is not. But containing zero is not enough: the set \(\{(x,y): x\ge 0\}\) contains it and still fails when multiplying by \(-1\).

Typical examples are lines and planes through the origin in \(\mathbb{R}^n\), the set \(\{0\}\) and \(V\) itself (the trivial subspaces), polynomials vanishing at a fixed point, and functions with \(f(0)=0\). In general, the solutions of a homogeneous equation form a subspace, while those of an equation with nonzero right-hand side do not.

Two frequent mistakes: believing that a union of subspaces is a subspace (the two axes of \(\mathbb{R}^2\) disprove it, since \((1,0)+(0,1)\) falls outside) and forgetting that the intersection always is. This concept prepares the next one: the span of some vectors is the smallest subspace containing them.

Think about it

Is \(\{(x,y,z): x+y+z=0\}\) a subspace of \(\mathbb{R}^3\)? And \(\{(x,y,z): x+y+z=0\ \text{or}\ x=0\}\)?

The first is: it contains \(0\) and the homogeneous equation is preserved by sums and scalar multiples. The second is not: \((1,-1,0)\) and \((0,1,1)\) are in it, but their sum \((1,0,1)\) satisfies neither condition.

Can a subspace of a real vector space have exactly two elements?

No. If it contains \(v\ne 0\) it contains all multiples \(cv\), which are infinitely many; so a subspace is \(\{0\}\) or infinite.

Exercises 10

  1. Which of these sets is a subspace of \(\mathbb{R}^2\)?
  2. A subset \(W\) of a vector space \(V\) is a subspace if and only if it…
  3. Let \(W=\{(x,y,z): x+y+z=0\}\). Which of these vectors belongs to \(W\)?
  4. Why is \(\{(x,y): x+y=1\}\) not a subspace of \(\mathbb{R}^2\)?
  5. Which of these subsets of \(\mathbb{R}^3\) is NOT a subspace?
  6. Inside the space of functions \(f:\mathbb{R}\to\mathbb{R}\), which set is a subspace?
  7. Let \(U=\{(x,y,z): x=2y\}\) and \(W=\{(x,y,z): z=0\}\). What do the vectors of \(U\cap W\) look like?
  8. In the space of polynomials of degree at most 2, which set is a subspace?
  9. Let \(U\) be the \(x\)-axis and \(W\) the \(y\)-axis in \(\mathbb{R}^2\). Why is \(U\cup W\) not a subspace?
  10. If \(W\) is a subspace of \(V\) and \(W\) contains a vector \(v\ne 0\), how many elements does \(W\) have?

References

  • Gilbert Strang, Introduction to Linear Algebra, 6th edition, Wellesley-Cambridge Press, 2023. §3.1
  • Sheldon Axler, Linear Algebra Done Right, 4th edition, Springer, 2024. §1.C
  • David C. Lay, Steven R. Lay, Judi J. McDonald, Linear Algebra and Its Applications, 6th edition, Pearson, 2021. §4.1