Previously, we saw a number of properties of the set \(N(A)\) where \(A\) is an \(m \times n\) matrix with entries from some field \(\mathbb{F}\).

In particular, the following the properties were highlighted:

1. For all \(u,v \in N(A)\), we have \(u + v \in N(A)\).

2. For all \(u \in N(A)\) and all \(\alpha \in \mathbb{F}\), we have \(\alpha u \in N(A)\).

3. There exist \(t_1,\ldots,t_k \in N(A)\) so that \(N(A) = \{ \alpha_1 t_1 + \cdots \alpha_k t_k : \alpha_1,\ldots,\alpha_k \in \mathbb{F}\}\).

Let \(P_2\) denote the set of polynomials in \(x\) with coefficients in \(\mathbb{F}\) having degree at most \(2\). In other words, \(P_2 = \{ ax^2 + bx + c : a,b,c \in \mathbb{F}\}\).

Replacing \(N(A)\) with \(P_2\) in the three properties above gives

1. For all \(u,v \in P_2\), we have \(u + v \in P_2\).

2. For all \(u \in P_2\) and all \(\alpha \in \mathbb{F}\), we have \(\alpha u \in P_2\).

3. There exist \(t_1,\ldots,t_k \in P_2\) so that \(P_2 = \{ \alpha_1 t_1 + \cdots \alpha_k t_k : \alpha_1,\ldots,\alpha_k \in \mathbb{F}\}\).

To see that property 2 holds, note that multiplying \(ax^2+bx+c\) by some \(\gamma \in \mathbb{F}\) gives \((\gamma a)x^2 + (\gamma b)x +\gamma c\), which is in \(P_2\).

To see that property 3 holds, let \(t_1 = x^2 + 0 x + 0\), \(t_2 = 0 x^2 + x + 0\), \(t_3 = 0 x^2 + 0 x + 1\). Then \(P_2 = \{ a t_1 + b t_2 + c t_3 : a,b,c \in \mathbb{F}\}\).

That \(N(A)\) and \(P_2\) shared the three properties is not a coincidence because both turn out to be examples of finite-dimensional vector spaces. Before we look at some more examples, let us look at the actual definition of a vector space.

Let \(V\) be a set equipped with addition and scalar multiplication with scalars from a field \(\mathbb{F}\). Suppose that \(V\) is closed under addition and scalar multiplication. The pair \(V,\mathbb{F}\) is called a vector space if

• \(x+(y+z)=(x+y)+z\) for all \(x, y,z \in V\)

• \(x+y=y+x\) for all \(x,y \in V\)

• \(\alpha(x+y)=\alpha x+ \alpha y\) for all \(x,y\in V\) and \(\alpha \in \mathbb{F}\)

• \((\alpha+\beta)x=\alpha x+ \beta x\) for all \(x \in V\) and \(\alpha, \beta\in\mathbb{F}\)

• there exists an element \(0_V \in V\), called the zero vector such that \(0_V+v = v\) for all \(v \in V\),

• for every \(v \in V\), there exists an element \(v' \in V\), called the additive inverse of \(v\), such that \(v + v' = 0_V\). One often writes \(v'\) as \(-v\).

• \(\alpha(\beta x)=(\alpha\beta) x\) for all \(x \in V\) and \(\alpha, \beta\in\mathbb{F}\)

• \(1x=x\) for all \(x \in V\) where 1 denotes the multiplicative identity in \(\mathbb{F}\).

\(N(A)\) is commonly called the nullspace of \(A\).

Remarks. In \(N(A)\), the zero vector is the \(m\)-tuple of 0's. In \(P_2\), it is the polynomial that is identically \(0\). In \(P_2\), the additive inverse of \(ax^2 + bx + c\) is \((-a)x^2 + (-b)x + (-c)\). One can then check that both \(N(A)\) and \(P_2\) satisfy all the above properties.

Subtraction of vectors

Subtraction of vectors \(x\) and \(y\), denoted by \(x-y\), is defined as \(x\) plus the additive inverse of \(y\). (One can show that \(x-y = x + (-1)y\) where \(-1\) is the additive inverse of the multiplicative identiy in \(\mathbb{F}\).)

Vector spaces are rather rich structures. There are a lot that can be said about them. We will see some of the notions that come out of the study of vector spaces.

Quick Quiz

Exercise

Let \(V\) denote the set of complex \(2 \times 2\) matrices. Show that \(V\) form a vector space under matrix addition and scalar multiplication.