- [[matrix]] # Idea A single number that reflects the **commonalities** (or similarities) between two mathematical objects (e.g., vectors, matrices, tensors, signals, images). The dot product of two vectors ${a}=\left[a_{1}, b_{a}, \ldots, b_{n}\right]$ and ${b}=\left[b_{1}, b_{2}, \ldots, b_{n}\right]$ is defined as: $ \boldsymbol{a} \cdot \boldsymbol{b}=\sum_{i}^{n} a_i b_i $ $\mathbf{a} \cdot \mathbf{b}=\sum_{i=1}^{n} a_{i} b_{i}=a_{1} b_{1}+a_{2} b_{2}+\cdots+a_{n} b_{n}$ $ \boldsymbol{a} \cdot \boldsymbol{b}=\left[\begin{array}{l} a_0 \\ a_1 \\ a_2 \\ a_3 \end{array}\right]\left[\begin{array}{l} b_0 \\ b_1 \\ b_2 \\ b_3 \end{array}\right]=\left[a_0 b_0+a_1 b_1+a_2 b_2+a_3 b_3\right]=c $ It's one of the most fundamental vector/matrix operations that underlies most computations in linear algebra. Examples: - [[variance-covariance matrix]] - [[linear combinations]] - [[matrix multiplication]] Note that [[dot product is covariance]]. ![[C1_W2_Lab04_dot_notrans.gif]] # References - [Dot product - Wikipedia](https://en.wikipedia.org/wiki/Dot_product)