Vector Space Structure of \mathbf{R}^{n} and \mathbf{C}^{n}

Real and Complex Vectors

An n-dimensional real vector is an n-tuple (x_1,\ldots, x_n) where the x_i are real numbers. If the x_i are complex instead of real, we have an n-dimensional complex vector.

We write \mathbf{R}^{n} and \mathbf{C}^{n} for the sets of n-dimensional real and complex vectors respectively.

One can visualize an element x=(x_1,\ldots,x_n) of \mathbf{R}^{n} as an “arrow” with its base at the origin and its endpoint at the point (x_1,\ldots, x_n) in n-dimensional Euclidean space.

Elements of \mathbf{R} or \mathbf{C} are often called scalars.

Operations

The set \mathbf{R}^{n} allows for:

Addition of vectors. If

x=(x_1,\ldots, x_n)\in R^{n} (or \mathbf{C}^{n}) and

y=(y_1,\ldots, y_n)\in\mathbf{R}^{n} (or \mathbf{C}^{n})

then x+y=(x_1+y_1,\dots, x_n+y_n) is in \mathbf{R}^{n} (or \mathbf{C}^{n}).

Multiplication of a vector by a scalar. If a\in\mathbf{R}\ \mathrm{and}\ y=(y_1,\ldots, y_n)\in\mathbf{R}^{n}

then ay=(ay_1,\ldots, ay_n). Similarly if a\in\mathbf{C} and y\in\mathbf{C}^{n}.

Axioms

These operations satisfy a list of axioms. For simplicity let’s work with \mathbf{R} and \mathbf{R}^{n} but the same things are true for \mathbf{R}^{n} and \mathbf{C}^{n}.

  1. Vector addition is commutative, associative, and if e=(0,\ldots,0) then x+e=e+x=x.
  2. Scalar multiplication distributes over addition so that (a+b)x=ax+bx and a(x+y)=ax+ay.
  3. Scalar multiplication is “associative” in the sense that a(bx)=(ab)x.
  4. 1x=x and 0x=e.

Real (and Complex) Vector spaces

Definition: A vector space over \mathbf{R} is a set V together with two operations: - Addition. Given x,y in V, there is an element x+y\in V. - Scalar Multiplication. Given a\in\mathbf{R} and x\in V there is an element ax\in V.

The two operations must satisfy the axioms listed above. In particular there must be an element e\in V so that x+e=e+x=x for all x\in V.

A vector space over \mathbf{C} is the same except that scalar multiplication is defined for a\in \mathbf{C}, not just \mathbf{R}.

The spaces \mathbf{R}^{n} and \mathbf{C}^{n} above are examples of real and complex vector spaces respectively.

Subspaces

Definition: A subspace W of a vector space V (for either \mathbf{R} or \mathbf{C}) is a subset of V that is a vector space under the inherited operations from V.

Example: Consider the set W of vectors (x_1,x_2,0) inside \mathbf{R}^{3}. This is a copy of \mathbf{R}^{2} sitting inside \mathbf{R}^{3} so it’s a subspace.

Example: Let W be the set of vectors in \mathbf{R}^{n} that satisfy x_1+x_2+\cdots+x_n=0. This is a subspace. Check: - If u=(u_1,\ldots, u_n) and v=(v_1,\ldots, v_n) are in W, then u+v is in W because

(u_1+v_1)+\cdots+(u_n+v_n)=(u_1+\cdots u_n)+(v_1+\cdots v_n)=0

Similarly au_1+au_2+\cdots+au_n=a(u_1+\cdots u_n)=0 so au is in W.

Finally e=(0,\ldots,0) is in W.

The other axioms hold for elements of W since all elements of W are also elements of V.

Example Let W be the set of vectors in \mathbf{R}^{n} that satisfy x_1+\cdots+x_n=1. This is NOT a subspace.

Proposition: A subset W of V is a subspace provided that it contains the element e and whenever u,v\in W it follows that u+v\in W and au\in W for all a\in\mathbf{R}.

You can replace \mathbf{R} by \mathbf{C} in the above to obtain the same results for complex vector spaces.

Four Crucial Definitions

Span of a set of vectors

Definition: Let V be a vector space and let S=\{v_1,\ldots, v_k\} be a set of vectors in V. The span of S is defined as either of the following equivalent things:

  1. The set of all vectors \sum_{i=1}^{n} a_i v_i as (a_1,\ldots, a_n) runs over \mathbf{R}^{n}.

  2. The intersection of all subspaces of V that contain S.

The set H in 1 is a subspace that contains S. If W is the intersection of all subspaces containing S, then W\subset H since H is among the sets being intersected. On the other hand, since S is contained in W, and W is a subspace, the elements in H are also in W. So W=H.

Linear Independence

Definition: A set of vectors v_1,\ldots, v_n in a vector space V is linearly independent if the only solution (a_1,\ldots, a_n) to the equation

\sum_{i=1}^{n} a_i v_i=0

is (0,\ldots, 0).

Equivalently:

  • a set S=\{v_1,\ldots, v_n\} is linearly independent if the span of any proper subset of S is a proper subspace of the span of S.

Suppose \sum a_{i}v_{i}=0 and that a_{j}\not=0. Then you can “solve” for v_{j} in terms of the other vectors: v_{j}=-\sum_{i\not=j} a_{i}/a_{j} v_{i} so v_{j} is in the span of the remaining v_{i}. Thus the span of the set S\backslash\{v_{j}\} is the same as the span of S.

Conversely, if there is a proper subset U of S whose span equals V, choose v_{i}\not\in U. Since v_{i} is in the span of U, you can write v_{i} = \sum_{v_{j}\in U} b_{j}v_{j} and therefore -v_{i}+\sum b_{j}v_{j}=0 and so S is not linearly independent.

  • a set S=\{v_1,\ldots, v_n\} is not linearly independent (is dependent) if and only if there is some 1\le j\le n such that v_{j}=\sum_{\substack{i=1 \\ i \neq j}}^{n} a_{i}v_{i}

This is implicit in the argument above.

Basis

A set S of vectors in a vector space V is called a basis of V if \mathrm{Span}(S)=V and S is linearly independent.

Dimension

Theorem: Every vector space has a basis.

Proof: This requires Zorn’s Lemma. If S_{1}\subset S_{2}\subset\cdots is an ascending sequence of linearly independent sets, their union is also linearly independent (because any linear relation would only involve finitely many vectors so would hold at some finite level of the chain). Therefore the linearly independent subsets of V form a partially ordered set in which every ascending chain has a maximal element. It follows that there is a maximal linearly independent set B in V.

Now choose any v\in V. If v is not in the span of B, then B\cap \{v\} would be linearly independent, contradicting maximality of B.

Hence B spans V and is therefore a basis.

Theorem: Any two bases of a vector space V have the same cardinality.

Proof: Suppose that X and Y are bases of the same vector space V. Each x\in X is a finite linear combination of elements of Y. This means there is a surjective finite-to-one map from Y to X. The surjectivity tells us that |Y|\ge |X|. On the other hand, a theorem in cardinal arithmetic (Hessenberg’s Theorem) says that if X is infinite and Y\to X is finite-to-one then |Y|\le |X|. Then The Cantor-Bernstein theorem says that X and Y have the same cardinality. ) We will establish the fact that any two finite bases of V have the same number of elements a little later.

Definition: The dimension of V is the cardinality of any basis of V.

If V has a finite basis, it is said to be finite-dimensional; otherwise it is infinite-dimensional.