Vector spaces · Definition

Basis and dimension

linear algebravector spaces

Before you start you need to know

A basis is the shortest list of vectors that describes an entire space; its length measures the space.

Definition

A basis \(\mathcal B\) of a vector space \(V\) is a linearly independent set that spans it; its number of vectors is the dimension of \(V\).

\[\mathcal{B}=\{v_1,\dots,v_n\},\quad \dim V=n\]

A basis does two things at once: it spans (every vector is reached) and it is independent (nothing is redundant). That is why each vector is written in exactly one way as \(v=a_1v_1+\cdots+a_nv_n\); the numbers \(a_i\) are its coordinates in that basis. The standard basis of \(\mathbb{R}^n\) is \(e_1,\dots,e_n\), where \(e_i\) has a 1 in position \(i\) and zeros elsewhere.

A space has many bases, but all of them have the same number of vectors; that number is the dimension. So \(\dim\mathbb{R}^n=n\), polynomials of degree at most \(3\) have dimension \(4\) (basis \(1,x,x^2,x^3\)) and \(2\times 3\) matrices have dimension \(6\). The space \(\{0\}\) has dimension \(0\).

Dimension simplifies checks: in a space of dimension \(n\), a set of \(n\) vectors is a basis if it is independent or if it spans; one of the two conditions is enough. Also, a subspace never has larger dimension than the space, and equals it only when it is the whole space. To find a basis of a span, discard the vectors that are combinations of the previous ones.

Typical mistake: confusing dimension with the number of vectors in an arbitrary set. \(\{(1,0),(2,0),(0,1)\}\) has three vectors, but it spans \(\mathbb{R}^2\) and its dimension is \(2\). The dimension of the column space of a matrix is its rank.

Think about it

Is \(\{(1,0,0),(0,1,0)\}\) a basis of \(\mathbb{R}^3\)?

No: it is independent, but it spans only the plane \(z=0\); a basis of \(\mathbb{R}^3\) needs 3 vectors.

Why is there no basis of \(\mathbb{R}^2\) with 3 vectors?

Three vectors of \(\mathbb{R}^2\) are always dependent, and a basis must be independent.

Exercises 10

  1. What is the dimension of \(\mathbb{R}^4\)?
  2. Which of these sets is a basis of \(\mathbb{R}^2\)?
  3. How many vectors does any basis of \(\mathbb{R}^3\) have?
  4. Write \((5,1)=a(1,1)+b(1,-1)\). What are \(a\) and \(b\)?
  5. What is the dimension of the plane \(x+y+z=0\) in \(\mathbb{R}^3\)?
  6. What is the dimension of the span of \((1,2,3)\), \((2,4,6)\) and \((0,1,0)\)?
  7. What is the dimension of the space of polynomials of degree at most 3?
  8. What is the dimension of the space of \(2\times 3\) matrices?
  9. If \(\dim V=n\) and \(S\) is a linearly independent set of \(n\) vectors of \(V\), then:
  10. If \(W\) is a subspace of a finite-dimensional space \(V\), then:

References

  • Gilbert Strang, Introduction to Linear Algebra, 6th edition, Wellesley-Cambridge Press, 2023. §3.4
  • Sheldon Axler, Linear Algebra Done Right, 4th edition, Springer, 2024. §2.C
  • David C. Lay, Steven R. Lay, Judi J. McDonald, Linear Algebra and Its Applications, 6th edition, Pearson, 2021. §4.3 y §4.5