6. Adding Vectors

If you are not familiar with vector addition, try to invent that for yourself. How would you define the addition of two vectors? Keep in mind that the result of adding two vectors should be another vector of the same size (or number of elements).

Vectors are ordered set of numbers. We denote them in columns or rows and call them column vectors and row vectors.

$$ \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix} ,\quad \begin{pmatrix} x_1, x_2, \cdots, x_n \end{pmatrix} $$

Addition of two vectors is defined as another vector whose components are the result of adding each component of the first vector to the corresponding component in the second vector. Consider two following two vectors of size $n$ where each component is a number (real or complex)

$$ \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix} + \begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix} = \begin{pmatrix} x_1 + y_1 \\ x_2 + y_2 \\ \vdots \\ x_n + y_n
\end{pmatrix}. $$

Therefore, we have $$ \begin{pmatrix} -1 \\ 3 \\ 0 \\ \sqrt{2} \end{pmatrix} + \begin{pmatrix} 2 \\ -1 \\ -\sqrt{2} \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ -\sqrt{2} \\ \sqrt{2} \end{pmatrix}. $$

Column vectors can be added to other column vectors only. Row vectors can be added to other row vectors only.

Zero vector

A zero vector $0$ is a special vector in a vector space which for any vector $|v\rangle$ has the following property

$$ |v\rangle + 0 = |v\rangle. $$