Bases¹

Definition. Let $V$ be a vector space over a field $F$. A list of vectors $v_1,\ldots, v_k \in V$ is a basis for $V$ if $v_1,\ldots, v_k$ are linearly independent and span $V$.

Theorem. The following conditions are equivalent, for $v_1,\ldots, v_n$ in a vector space $V$.

(1) $v_1,\ldots, v_n$ form a basis for $V$.

(2) Every element of $V$ can be written uniquely as a linear combination of $v_1,\ldots, v_n$.

(3) There is an isomorphism of $F^n$ with $V$ such that the standard basis vectors $e_1,\ldots, e_n$ of $F^n$ correspond to $v_1,\ldots, v_n \in V$.

(4) $v_1,\ldots, v_n$ are a minimal spanning set, in the sense that if any one of them is omitted then the remaining vectors do not span $V$.

(5) $v_1,\ldots, v_n$ is a maximal linearly independent set, in the sense that if any vector in $V$ is added to the list, the new list is linearly dependent.

Lemma. If $V$ has a basis with $n$ elements, then any $n+1$ elements of $V$ are linearly dependent.

Theorem. If $V$ has a basis with $n$ elements, then any other basis also has $n$ elements.

Definition. If $V$ has a basis with $n$ elements, we say $V$ has dimension $n$ over $F$.

Notes. (a) By the Theorem, there is no ambiguity about what the dimension is. (b) Some vector spaces, such as the vector space $\mathcal C(\mathbb{R}\rightarrow \mathbb{R})$ of continuous functions, are infinite dimensional; there is no finite basis.

Useful observation: In a linearly dependent list of vectors, some vector is in the span of the preceding vectors in the list. Or to put it the other way around, if a list of vectors has the property that no member is in the span of the preceding ones, it is linearly independent.

(What does ``the span of the preceding ones'' mean in the case of the first vector in the list?)

Developing Intuition

If you develop intuition based on $\mathbb{R}^3$, it should be a good guide to what is true about finite-dimensional vector spaces in general (except for facts that depend on the dimension of the whole space being 3!)

Try these. Where one is false, think of a counterexample. Where one is true, see if you can prove it.

True or false?

1. A subspace of a finite-dimensional vector space is always finite-dimensional.

2. For three vectors $v_1, v_2, v_3$ in a vector space $V$, if each two are linearly independent, then the three are linearly independent.

3. If $v_1,\ldots, v_n$ span $V$, then some subset of them is a basis for $V$.

4. If $v_1,\ldots, v_n$ are a basis of $V$ and $W$ is a subspace of $V$, then some subset of $\{v_1,\ldots, v_n\}$ is a basis for $W$.

5. If $w \in$   Span$(v_1,..,v_n)$, then Span$(v_1,\ldots, v_n,w) =$   Span$(v_1,..,v_n)$.

6. If $v_1,\ldots, v_k$ are linearly independent in a finite-dimensional vector space $V$, then this list can be extended to a basis.

7. Any linearly independent list of vectors is a basis for the subspace they span.

8. If $W_1, W_2$ are subspaces of a finite-dimensional vector space $V$, then dim$W_1 +$   dim$W_2 \leq$   dim$V$.

9. If Span$(v_1) =$   Span$(v_2)$ then $v_1 = r v_2$ for some scalar $r$.

10. For any two 2-dimensional subspaces of a vector space, their intersection contains a nonzero vector.