4.1.1. Row Vector

A set of n real numbers 𝑢𝑖 (i = 1, 2, 3, …, n) is written in the following form:
• 𝑢 = 𝑢₁, 𝑢₂, …, 𝑢ₙ
This is called an n-dimensional row vector of real numbers. In our work, we will use this notation only in cases of uncertainty.

4.1.2. Column Vector
Similarly, a set of n real numbers 𝑣𝑖 (i = 1, 2, 3, …, n) written in the following form:
𝑣₁
𝑣₂
𝑣ₙ
is called an n-dimensional column vector of real numbers.

For example:
1
2
3

This vector is a column vector in ℝ³ because it consists of 3 real numbers.

In quantum computing, the elements of row and column vectors are typically complex numbers. This is a more common situation in quantum computing.

For example:
• 𝑐 = ( 2 + i, 5 + 3i)

The vectors c defined above are vectors in ℂ² because each element of these vectors is a complex number.

4.1.3. Scalar Concept
Each element of a vector in ℝⁿ is considered a scalar. Similarly, when working with vectors in ℂⁿ, the term “scalar” refers to complex numbers, i.e., it represents each element of ℂ. Therefore, when the elements of a vector are complex numbers, these elements are also considered complex scalars.
If every element of a vector in ℝⁿ or ℂⁿ is the scalar 0, then this vector is considered the zero vector.

4.1.4. Basic Set in Quantum Computing
The set B = {0, 1} can be considered as the basic set.

𝑥 = (1, 0, 0, 1)

and 𝑦 =
0
1

Here, the vectors 𝑥 and 𝑦 are elements of 𝐵⁴ and 𝐵², respectively. These vectors represent binary values used in quantum computing.