5.1 Inner Product Between Vectors

#Type/Definition

Types: N/A
Examples: 5.1.1 Final grade calculations using inner product
Constructions: 5.3 The 2-norm of a Vector, 5.9 Orthogonal Vectors
Generalizations: N/A

Properties: 5.2 Algebraic Properties of Inner Products, 5.7 Cosine Formula for Inner Product
Sufficiencies: N/A
Questions: N/A

Inner Product Between Vectors

Let $n \in N$ and suppose that $\vec{x}, \vec{y} \in R^{n}$ . Then, the inner product between $\vec{x}$ and $\vec{y}$ is given by

{\vec{x}}_{n \times 1} \cdot {\vec{y}}_{n \times 1} = {[\begin{matrix} x_{1} \\ x_{2} \\ ⋮ \\ x_{n} \end{matrix}]}_{n \times 1} \cdot {[\begin{matrix} y_{1} \\ y_{2} \\ ⋮ \\ y_{n} \end{matrix}]}_{n \times 1} = x_{1} \cdot y_{1} + x_{2} \cdot y_{2} + \dots + x_{n} y_{n} = \sum_{j = 1}^{n} x_{j} y_{j} .

Remark. Notice that the dot product is a function with domain $R^{n} \times R^{n}$ and codomain $R$ . Therefore, the output of the dot product between any two vectors is a real number.
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