Introduction to Linear Algebra - RPIQuantumComputing/QuantumCircuits GitHub Wiki

Linear algebra is essential for quantum computing. Linear algebra is a branch of mathematics that deals with vector spaces and linear mappings between these spaces.

Vectors: The basis of linear algebra is a vector. Geometrically, vectors can be thought of as directed line segments, where the direction indicates the line along which the vector extends, and the length represents its magnitude.

Linear Transformations: A linear transformation is a function between two vector spaces that preserves vector addition and scalar multiplication. In other words, it satisfies two properties: additivity (f(u + v) = f(u) + f(v)) and homogeneity (f(k * v) = k * f(v)), where u and v are vectors, k is a scalar, and f is the transformation. Linear transformations can be represented by matrices, and they include operations like rotations, translations, reflections, and scaling.

Matrix multiplication: Matrix multiplication is a fundamental operation in linear algebra. Given two matrices A and B, their product AB is computed by taking the dot product of the rows of matrix A with the columns of matrix B. The resulting matrix has dimensions determined by the number of rows of A and the number of columns of B. Matrix multiplication is associative but not commutative, meaning that the order of multiplication matters.

Eigenvalues: In linear algebra, an eigenvalue is a scalar that represents how a linear transformation (represented by a matrix) stretches or compresses a vector. An eigenvector is a non-zero vector that remains in the same direction after the transformation, only scaled by the eigenvalue. Eigenvalues and eigenvectors are crucial in various applications, such as in physics (quantum mechanics), engineering (vibration analysis), and data analysis (dimensionality reduction).

Unitary Matrix: In linear algebra, a unitary matrix is a complex square matrix that satisfies the following property: ( U^{}U = UU^{} = I ) where U^{*} denotes the conjugate transpose of U and I is the identity matrix. In other words, a matrix U is unitary if its conjugate transpose is its inverse.