1.0. Matrix multiplication and dot products

Notice that a row vector times a column vector is really a dot product: If we write all vectors as column vectors, we have $v \cdot w = v ^ t w$. (Here we treat a $1 \times 1$ matrix as a scalar.)

A matrix product $C = AB$ is really a matrix of dot products:

$C _ {ij}$ is the dot product of row $i$ of $A$ with column $j$ of $B$.

Problem V-1. Suppose the rows of an $n \times n$ matrix $P$ are orthonormal, meaning that they are all of length 1 and that any two are orthogonal (perpendicular). Show that $P P ^ t = I$.

(Method: The length of a vector $v$ is $\sqrt{v \cdot v}$, so $v$ has length 1 when $v \cdot v = 1$. Also, two vectors are perpendicular when their dot product is 0.)