Bases

Definition 1. A (linear) basis (or a coordinate system ) in a vector space $𝒱$ is a set $𝒳$ of linearly independent vectors such that every vector in $𝒱$ is a linear combination of elements of $𝒳$ . A vector space $𝒱$ is finite-dimensional if it has a finite basis.

Except for the occasional consideration of examples we shall restrict our attention, throughout this book, to finite-dimensional vector spaces.

For examples of bases we turn again to the spaces $𝒫$ and $ℂ^{n}$ . In $𝒫$ , the set ${x_{n}}$ , where $x_{n} (t) = t^{n}$ , $n = 0, 1, 2, \dots$ , is a basis; every polynomial is, by definition, a linear combination of a finite number of $x_{n}$ . Moreover $𝒫$ has no finite basis, for, given any finite set of polynomials, we can find a polynomial of higher degree than any of them; this latter polynomial is obviously not a linear combination of the former ones.

An example of a basis in $ℂ^{n}$ is the set of vectors $x_{i}$ , $i = 1, \dots, n$ , defined by the condition that the $j$ -th coordinate of $x_{i}$ is $δ_{i j}$ . (Here we use for the first time the popular Kronecker $δ$ ; it is defined by $δ_{i j} = 1$ if $i = j$ and $δ_{i j} = 0$ if $i \neq j$ .) Thus we assert that in $ℂ^{3}$ the vectors $x_{1} = (1, 0, 0)$ , $x_{2} = (0, 1, 0)$ , and $x_{3} = (0, 0, 1)$ form a basis. It is easy to see that they are linearly independent; the formula $x = (ξ_{1}, ξ_{2}, ξ_{3}) = ξ_{1} x_{1} + ξ_{2} x_{2} + ξ_{3} x_{3}$ proves that every $x$ in $ℂ^{3}$ is a linear combination of them.

In a general finite-dimensional vector space $𝒱$ , with basis ${x_{1}, \dots, x_{n}}$ , we know that every $x$ can be written in the form $x = \sum_{i} ξ_{i} x_{i};$ we assert that the $ξ$ ’s are uniquely determined by $x$ . The proof of this assertion is an argument often used in the theory of linear dependence. If we had $x = \sum_{i} η_{i} x_{i}$ , then we should have, by subtraction, $\sum_{i} (ξ_{i} - η_{i}) x_{i} = 0.$ Since the $x_{i}$ are linearly independent, this implies that $ξ_{i} - η_{i} = 0$ for $i = 1, \dots, n$ ; in other words, the $ξ$ ’s are the same as the $η$ ’s. (Observe that writing ${x_{1}, \dots, x_{n}}$ for a basis with $n$ elements is not the proper thing to do in case $n = 0$ . We shall, nevertheless, frequently use this notation. Whenever that is done, it is, in principle, necessary to adjoin a separate discussion designed to cover the vector space $𝒪$ . In fact, however, everything about that space is so trivial that the details are not worth writing down, and we shall omit them.)

Theorem 1. If $𝒱$ is a finite-dimensional vector space and if ${y_{1}, \dots, y_{m}}$ is any set of linearly independent vectors in $𝒱$ , then, unless the $y$ ’s already form a basis, we can find vectors $y_{m + 1}, \dots, y_{m + p}$ so that the totality of the $y$ ’s, that is, ${y_{1}, \dots, y_{m}, y_{m + 1}, \dots, y_{m + p}}$ , is a basis. In other words, every linearly independent set can be extended to a basis.

Proof. Since $𝒱$ is finite-dimensional, it has a finite basis, say ${x_{1}, \dots, x_{n}}$ . We consider the set $𝒮$ of vectors $y_{1}, \dots, y_{m}, x_{1}, \dots, x_{n}$ in this order, and we apply to this set the theorem of Section: Linear combinations several times in succession. In the first place, the set $𝒮$ is linearly dependent, since the $y$ ’s are (as are all vectors) linear combinations of the $x$ ’s. Hence some vector of $𝒮$ is a linear combination of the preceding ones; let $z$ be the first such vector. Then $z$ is different from any $y_{i}$ , $i = 1, \dots, m$ (since the $y$ ’s are linearly independent), so that $z$ is equal to some $x$ , say $z = x_{i}$ . We consider the new set \mathcal{S}^{\prime} of vectors $y_{1}, \dots, y_{m}, x_{1}, \dots, x_{i - 1}, x_{i + 1}, \dots, x_{n} .$ We observe that every vector in $𝒱$ is a linear combination of vectors in \mathcal{S}^{\prime} , since by means of $y_{1}, \dots, y_{m}, x_{1}, \dots, x_{i - 1}$ we may express $x_{i}$ , and then by means of $x_{1}, \dots, x_{i - 1}, x_{i}, x_{i + 1}, \dots, x_{n}$ we may express any vector. (The $x$ ’s form a basis.) If \mathcal{S}^{\prime} is linearly independent, we are done. If it is not, we apply the theorem of Section: Linear combinations again and again the same way till we reach a linearly independent set containing $y_{1}, \dots, y_{m}$ , in terms of which we may express every vector in $𝒱$ . This last set is a basis containing the $y$ ’s. ◻

EXERCISES

Exercise 1.

Prove that the four vectors \begin{align} & x=(1,0,0) \\ & y=(0,1,0) \\ & z=(0,0,1) \\ & u=(1,1,1) \end{align}in $ℂ^{3}$ form a linearly dependent set, but any three of them are linearly independent. (To test the linear dependence of vectors $x = (ξ_{1}, ξ_{2}, ξ_{3}), y = (η_{1}, η_{2}, η_{3})$ , and $z = (ζ_{1}, ζ_{2}, ζ_{2})$ in $ℂ^{3}$ , proceed as follows. Assume that $α$ , $β$ , and $γ$ can be found so that $α x + β y + γ z = 0$ . This means that \begin{align} & \alpha \xi_{1}+\beta \eta_{1}+\gamma \xi_{1}=0 \\ & \alpha \xi_{2}+\beta \eta_{2}+\gamma \zeta_{2}=0 \\ & \alpha \xi_{3}+\beta \eta_{3}+\gamma \xi_{3}=0 \end{align}The vectors $x$ , $y$ , and $z$ are linearly dependent if and only if these equations have a solution other than $α = β = γ = 0$ .)
If the vectors $x$ , $y$ , $z$ , and $u$ in $𝒫$ are defined by $x (t) = 1$ , $y (t) = t$ , $z (t) = t^{2}$ , and $u (t) = 1 + t + t^{2}$ , prove that $x$ , $y$ , $z$ , and $u$ are linearly dependent, but any three of them are linearly independent.

Exercise 2. Prove that if $ℝ$ is considered as a rational vector space (see. Section: Examples , (8)), then a necessary and sufficient condition that the vectors $1$ and $ξ$ in $ℝ$ be linearly independent is that the real number $ξ$ be irrational.

Exercise 3. Is it true that if $x$ , $y$ , and $z$ are linearly independent vectors, then so also are $x + y$ , $y + z$ , and $z + x$ ?

Exercise 4.

Under what conditions on the scalar $ξ$ are the vectors $(1 + ξ, 1 - ξ)$ and $(1 - ξ, 1 + ξ)$ in $ℂ^{2}$ linearly dependent?
Under what conditions on the scalar $ξ$ are the vectors $(ξ, 1, 0)$ , $(1, ξ, 1)$ , and $(0, 1, ξ)$ in $ℝ^{2}$ linearly dependent?
What is the answer to (b) for $ℚ^{3}$ (in place of $ℝ^{3}$ )?

Exercise 5.

The vectors $(ξ_{1}, ξ_{2})$ and $(η_{1}, η_{2})$ in $ℂ^{2}$ are linearly dependent if and only if $ξ_{1} η_{2} = ξ_{2} η_{1}$ .
Find a similar necessary and sufficient condition for the linear dependence of two vectors in $ℂ^{2}$ . Do the same for three vectors in $ℂ^{3}$ .
Is there a set of three linearly independent vectors in $ℂ^{2}$ ?

Exercise 6.

Under what conditions on the scalars $ξ$ and $η$ are the vectors $(1, ξ)$ and $(1, η)$ in $ℂ^{2}$ linearly dependent?
Under what conditions on the scalars $ξ$ , $η$ , and $ζ$ are the vectors $(1, ξ, ξ^{2})$ , $(1, η, η^{2})$ , and $(1, ζ, ζ^{2})$ in $ℂ^{2}$ linearly dependent?
Guess and prove a generalization of (a) and (b) to $ℂ^{n}$ .

Exercise 7.

Find two bases in $ℂ^{4}$ such that the only vectors common to both are $(0, 0, 1, 1)$ and $(1, 1, 0, 0)$ .
Find two bases in $ℂ^{4}$ that have no vectors in common so that one of them contains the vectors $(1, 0, 0, 0)$ and $(1, 1, 0, 0)$ and the other one contains the vectors $(1, 1, 1, 0)$ and $(1, 1, 1, 1)$ .

Exercise 8.

Under what conditions on the scalar $ξ$ do the vectors $(1, 1, 1)$ and $(1, ξ, ξ^{2})$ form a basis of $ℂ^{3}$ ?
Under what conditions on the scalar $ξ$ do the vectors $(0, 1, ξ)$ , $(ξ, 0, 1)$ , and $(ξ, 1, 1 + ξ)$ form a basis of $ℂ^{3}$ ?

Exercise 9. Consider the set of all those vectors in $ℂ^{2}$ each of whose coordinates is either $0$ or $1$ ; how many different bases does this set contain?

Exercise 10. If $𝒳$ is the set consisting of the six vectors $(1, 1, 0, 0)$ , $(1, 0, 1, 0)$ , $(1, 0, 0, 1)$ , $(0, 1, 1, 0)$ , $(0, 1, 0, 1)$ , $(0, 0, 1, 1)$ in $ℂ^{4}$ , find two different maximal linearly independent subsets of $𝒳$ . (A maximal linearly independent subset of $𝒳$ is a linearly independent subset $𝒴$ of $𝒳$ that becomes linearly dependent every time that a vector of $𝒳$ that is not already in $𝒴$ is adjoined to $𝒴$ .)

Exercise 11. Prove that every vector space has a basis. (The proof of this fact is out of reach for those not acquainted with some transfinite trickery, such as well-ordering or Zorn’s lemma.)

EXERCISES

Quick Links

Social Media