0.2 Vector spaces

Following Peano, we define a vector space as a mathematical structure in which the notion of linear combination “makes sense”.

More formally, a complex vector space is a set V such that, given any two vectors a and b (that is, any two elements of V) and any two complex numbers \alpha and \beta, we can form the linear combination2 \alpha a+\beta b, which is also a vector in V.

A subspace of V is any subset of V which is closed under vector addition and multiplication by complex numbers. Here we start using the Dirac bra-ket notation and write vectors in a somewhat fancy way as |\text{label}\rangle, where the “label” is anything that serves to specify what the vector is. For example, |\uparrow\rangle and |\downarrow\rangle may refer to an electron with spin up or down along some prescribed direction and |0\rangle and |1\rangle may describe a quantum bit (a “qubit”) holding either logical 0 or 1. These are often called ket vectors, or simply kets. (We will deal with “bras” in a moment). A basis in V is a collection of vectors |e_1\rangle,|e_2\rangle,\ldots,|e_n\rangle such that every vector |v\rangle in V can be written (in exactly one way) as a linear combination of the basis vectors; |v\rangle=\sum_i v_i|e_i\rangle. The number of elements in a basis is called the dimension of V. (Showing that this definition is independent of the basis that we choose is a “fun” linear algebra exercise). The most common n-dimensional complex vector space is the space of ordered n-tuples of complex numbers, usually written as column vectors: \begin{gathered} |a\rangle = \begin{bmatrix}a_1\\a_2\\\vdots\\a_n\end{bmatrix} \qquad |b\rangle = \begin{bmatrix}b_1\\b_2\\\vdots\\b_n\end{bmatrix} \\\alpha|a\rangle+\beta|b\rangle = \begin{bmatrix}\alpha a_1+\beta b_1\\\alpha a_2+\beta b_2\\\vdots\\\alpha a_n+\beta b_n\end{bmatrix} \end{gathered}

In fact, this is the space we will use most of the time. Throughout the course we will deal only with vector spaces of finite dimensions. This is sufficient for all our purposes and we will avoid many mathematical subtleties associated with infinite dimensional spaces, for which we would need to tools of functional analysis.

1. As we said, there are certain “nice properties” that these things must satisfy. Addition of vectors must be commutative and associative, with an identity (the zero vector, which will always be written as \mathbf{0} ) and an inverse for each v (written as -v). Multiplication by complex numbers must obey the two distributive laws: (\alpha+\beta)v = \alpha v+\beta v and \alpha (v+w) = \alpha v+\alpha w.↩︎