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 $$N(A) = \{ \alpha_1 t_1 + \cdots \alpha_k t_k : \alpha_1,\ldots,\alpha_k \in \mathbb{F}\}$$.

We now show that these properties are also enjoyed by a very different-looking set.

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 1 holds, note that adding $$a_1x^2 + b_1x +c_1$$ and $$a_2x^2 + b_2x +c_2$$ gives $$(a_1+a_2)x^2 + (b_1+b_2)x + (c_1+c_2)$$, which is in $$P_2$$

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 $$\mathbb{F}$$.

We call $$V$$ a vector space over the field $$\mathbb{F}$$. The elements of $$V$$ are called vectors.

$$N(A)$$ 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.

## 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.