Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum

Let $n\ge1$ be a natural number and let $\mathbb{R}$ be the real numbers. Let $P,Q,M$ be real $n\times n$ matrices, with entries $P_{ij}$ and so on, let $z,z',w$ be points of Euclidean space $\mathbb{R}^n$, and let $\mu\in\mathbb{R}$.

Write $P+Q$ for the sum of matrices, $P-Q$ for their difference, $\mu P$ for the scalar multiple of a matrix, $I_n$ for the identity matrix of size $n$, and $Mz$ for the matrix-vector product. On $\mathbb{R}^n$, regarded as a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, write $z+z'$ for the sum, $\mu z$ for the scalar multiple, and $z-z'$ and $w\cdot z$ for the difference and the dot product.

Then the following hold.

1. (Linearity in the matrix)

$$(P+Q)z=Pz+Qz,\qquad (P-Q)z=Pz-Qz,\qquad (\mu P)z=\mu\,(Pz).$$

2. (Identity matrix)

$$I_nz=z .$$

3. (Linearity in the vector)

$$M(z+z')=Mz+Mz',\qquad M(z-z')=Mz-Mz',\qquad M(\mu z)=\mu\,(Mz).$$

4. (Quadratic form as a double sum)

$$w\cdot(Mz)=\sum_{i=1}^{n}\ \sum_{j=1}^{n}M_{ij}\,w_i\,z_j .$$