Row vectors, column vectors, and matrices

We define an ($n$-dimensional) (row) vector to be an element of ${\mathbb{R}}^n$ endowed with addition and scalar multiplication. 

So, a row vector is something like $(x_1,x_2,\cdots ,x_n)$. We may omit the commas as in $(x_1{x}_2\cdots x_n)$. But vectors are not just elements of ${\mathbb{R}}^n$. We define addition between row vectors by

$$\begin{array}{}\text{(eq:vadd)}& (x_1,\cdots ,x_n)+(y_1,\cdots ,y_n)=(x_1+y_1,\cdots ,x_n+y_n).\end{array}$$

We also define multiplication by a real number $\lambda \in \mathbb{R}$ by

$$\begin{array}{}\text{(eq:svmul)}& \lambda \cdot (x_1,\cdots ,x_n)=(\lambda x_1,\cdots ,\lambda x_n).\end{array}$$

In this context, the real numbers are called scalars because it scales the vector.

In summary, vectors are elements of ${\mathbb{R}}^n$ where addition and scalar multiplications are defined. We will give more characterization of vectors in later posts.

Example. $(1,2),(-5,3)\in {\mathbb{R}}^2$ are 2-dimensional row vectors. We can add them as

$$(1,2)+(-5,3)=(1-5,2+3)=(-4,5).$$

We can multiply them by scalars (real numbers):

$$\begin{array}{rcl}\pi \cdot (1,2)& =& (\pi ,2\pi ),\\ -1\cdot (-5,3)& =& (5,-3).\end{array}$$

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Instead of writing $n$ real numbers in a row like $(x_1,\cdots ,x_n)$, we could write them in a column like

$$\left(\begin{array}{c}{x}_1\\ ⋮\\ x_n\end{array}\right)$$

which we call a column vector. Addition and scalar multiplication can be defined similarly to row vectors.

If $n$ is not prime, we can write a point in ${\mathbb{R}}^n$ in multiple rows and columns. For example, a point in ${\mathbb{R}}^6$ may be written as

$$(x_1,x_2,x_3,x_4,x_5,x_6)$$

or

$$\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\\ x_5\\ x_6\end{array}\right)$$

or

$$\left(\begin{array}{ccc}{x}_1& x_2& x_3\\ x_4& x_5& x_6\end{array}\right)$$

or

$$\left(\begin{array}{cc}{x}_1& x_2\\ x_3& x_4\\ x_5& x_6\end{array}\right).$$

Such a rectangular array of numbers is called a matrix. If there are $m$ rows and $n$ columns in a matrix, we call it an $m\times n$ matrix and regard it as a member of the set ${\mathbb{R}}^{m\times n}$. We can also define element-wise addition and scalar multiplication for matrices. Now we can see that an $n$-dimensional row vector is really a $1\times n$ matrix, and an $n$-dimensional column vector is an $n\times 1$ matrix. Accordingly, we can define addition and scalar multiplication of matrices in the same way as vectors.

Example. 

$$\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right)+\left(\begin{array}{ccc}-3& -1& 2\\ 1& -2& 3\end{array}\right)=\left(\begin{array}{ccc}-2& 1& 5\\ 5& 3& 9\end{array}\right),$$

$$5\cdot \left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right)=\left(\begin{array}{ccc}5& 10& 15\\ 20& 25& 30\end{array}\right).$$

Note that two matrices can be added if and only if they have the same shape. □

Remark. So far, we have worked with ${\mathbb{R}}^n$ to define vectors (including matrices). We can similarly define vectors and matrices on ${\mathbb{C}}^n$ or ${\mathbb{Q}}^n$ where scalars are complex or rational numbers, respectively. Note that $\mathbb{R}$, $\mathbb{C}$ and $\mathbb{Q}$ are fields. In fact, we may define vectors and matrices over any field. □

Definition (Linear combination)

Let ${\mathbf{v}}_1,{\mathbf{v}}_2,\cdots ,{\mathbf{v}}_n$ be vectors and ${\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_n$ be scalars. The expression of the form

$${\lambda }_1{\mathbf{v}}_1+{\lambda }_2{\mathbf{v}}_2+\cdots +{\lambda }_n{\mathbf{v}}_n$$

is called the linear combination of the vectors ${\mathbf{v}}_1,{\mathbf{v}}_2,\cdots ,{\mathbf{v}}_n$ with the coefficients ${\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_n$.

Example. 

$$(5,-9)=3(1,-1)+2(1,-3).$$

Thus, $(5,-9)$ can be represented as a linear combination of $(1,-1)$ and $(1,-3)$ with coefficients 3 and 2, respectively. □