Quantum computing relies on linear algebra to describe the state of a system and how that state changes over time. While classical logic uses Boolean algebra, quantum mechanics uses vector spaces and matrices to predict outcomes. This chapter establishes the mathematical tools required to model qubit behavior and construct algorithms.

We begin by defining the role of complex numbers, denoted as $z = a + bi$, which act as the amplitudes for quantum states. You will learn to represent these states as vectors within a Hilbert space. The text explains how to perform operations with these vectors, specifically using the inner product to calculate the probability of measurement outcomes and the outer product to represent operators.

The focus then shifts to the operators themselves. You will work with unitary matrices, which are required because they preserve the total probability of a quantum system. We also define eigenvalues and eigenvectors, the mathematical components that correspond to the actual values observed during a measurement. For example, if an operator $A$ acts on a vector $v$ such that:

$Av = \lambda v$

the scalar $\lambda$ represents the measurable quantity. Finally, we apply these concepts programmatically. You will use the Python library NumPy to generate vectors and matrices, effectively building a simulator for the mathematical operations that control quantum hardware.