Vector spaces over ℂ

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To complete this section we extend our set of scalars from real numbers to complex numbers.

The complex numbers $\mathbb C$ are formed from the real numbers by adjoining $i = \sqrt {-1}$ , so that $i^2 = -1$ . Every complex number can b written uniquely as $z = a + bi$ where $a,b\in \mathbb R$ . Every complex number has both a real and imaginary part, defined as $Re(a+bi) = a,\qquad Im(a+bi) = b$ For $z = a + bi$ , we recall its conjugate is defined to be $\ov {z} := a - bi$ . The real numbers embed naturally in $\mathbb C$ as those whose imaginary component is zero: $\mathbb R\ni a\mapsto a + 0i\in \mathbb C$ . Alternatively $\mathbb R$ identifies with those $z\in \mathbb C$ satisfying $z = \ov {z}$ .

A vector space over $\mathbb C$ satisfies exactly the same axioms as a vector space over $\mathbb R$ , the one difference being that scalars are allowed to be complex. Beyond that, all of the constructions and definitions over $\mathbb R$ given above extend without change to working over $\mathbb C$ . In analogy to $\mathbb R^n$ we have $\mathbb C^n$ , which can be viewed as the complex vector space generated by the same standard basis vectors we had for $\mathbb R^n$ ; vectors in $\mathbb C^n$ are naturally represented by $n\times 1$ column vectors with entries in $\mathbb C$ : $\mathbb C^n := \{[z_1\ z_2\ \dots z_n]^T\ |\ z_i\in \mathbb C\}$ In cases which involve both real and complex scalars, one has to take care as to which set of numbers one is working over, because this will make a difference when computing quantities such as dimension. To illustrate, $\mathbb C^1$ is a vector space over $\mathbb C$ with dimension $1$ . On the other hand, because $\mathbb R\subset \mathbb C$ , we could also consider $\mathbb C^1$ as a vector space over $\mathbb R$ by restriction of scalars; in other words, by only allowing scalar multiplication by real numbers. Over $\mathbb R$ , $\mathbb C^1$ has dimension 2, not 1, with basis $\{[1], [i]\}$ . Because of this it is not unusual (when discussing dimension) to emphasize the base field when there is any possibility of confusion or abiguity. In general, for any finite dimensional vector space $V$ over $\mathbb C$ , if $Dim_{\mathbb C}(V) = n$ then $Dim_{\mathbb R}(V) = 2n$ .