Add content for vectors-matrix-ops/vectors-matrix-operations

Add actual content to the skeleton of the
`vectors-matrix-ops/vectors-matrix-operations` section.

Signed-off-by: Eggert Karl Hafsteinsson <[email protected]>
Signed-off-by: Teodor Dutu <[email protected]>
Signed-off-by: Razvan Deaconescu <[email protected]>

chapters/vectors-matrix-ops/vectors-matrix-operations/reading/read.md

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+# Vectors and Matrix Operations
+
+## Numbers, Vectors, Matrices
+
+Recall that the set of real numbers is $\mathbb{R}$ and that a vector, $v \in \mathbb{R}^n$, is just an $n$-tuple of numbers.
+Similarly, an $n \times m$ matrix is just a table of numbers, with $n$ rows and $m$ columns and we can write
+
+$$A_{mn} \in \mathbb{R}^{mn}$$
+
+Note that a vector is normally considered equivalent to a $n\times 1$ matrix i.e. we view these as column vectors.
+
+### Examples
+
+:::info Example
+
+In R, a vector can be generated with:
+
+```text
+> X <- 3:6
+> X
+[1] 3 4 5 6
+```
+
+A matrix can be generated in R as follows,
+
+```text
+> matrix(X)
+     [,1]
+[1,]    3
+[2,]    4
+[3,]    5
+[4,]    6
+```
+
+:::
+
+:::note Note
+
+We note that R distinguishes between vectors and matrices.
+
+:::
+
+## Elementary Operations
+
+We can define multiplication of a real number $k$ and a vector $v=(v_1,\ldots,v_n)$ by $k\cdot v=(kv_1,\ldots,kv_n)$.
+The sum of two vectors in $\mathbb{R}^n$, $v=(v_1,\ldots,v_n)$ and $u=(u_1,\ldots,u_n)$ is defined as the vector $v+u=(v_1+u_1,\ldots,v_n+u_n)$.
+We can define multiplication of a number and a matrix and the sum of two matrices (of the same sizes) similarly.
+
+### Examples
+
+:::info Example
+
+```text
+> A <- matrix(c(1,2,3,4), nr=2, nc=2)
+
+> A
+     [,1] [,2]
+[1,]    1    3
+[2,]    2    4
+
+> B <- matrix(c(1,0,2,1), nr=2, nc=2)
+
+> B
+     [,1] [,2]
+[1,]    1    2
+[2,]    0    1
+
+> A+B
+     [,1] [,2]
+[1,]    2    5
+[2,]    2    5
+```
+
+:::
+
+## The Tranpose of a Matrix
+
+In R, matrices may be constructed using the `matrix` function and the transpose of $A$, $A^\prime$, may be obtained in R by using the `t` function:
+
+```text
+> A <- matrix(1:6, nrow=3)
+
+> t(A)
+     [,1] [,2] [,3]
+[1,]    1    2    3
+[2,]    4    5    6
+```
+
+### Details
+
+If $A$ is an $n \times m$ matrix with element $a_{ij}$ in row $i$ and column $j$, then $A^\prime$ or $A^T$ is the $m\times n$ matrix with element $a_{ij}$ in row $j$ and column $i$.
+
+### Examples
+
+:::info Example
+
+Consider a vector in R
+
+```text
+> x <- 1:4
+
+> x
+[1] 1 2 3 4
+
+> t(x)
+     [,1] [,2] [,3] [,4]
+[1,]    1    2    3    4
+
+> matrix(x)
+     [,1]
+[1,]    1
+[2,]    2
+[3,]    3
+[4,]    4
+> t(matrix(x))
+     [,1] [,2] [,3] [,4]
+```
+
+:::
+
+:::note Note
+
+The first solution gives a $1 \times n$ matrix and the second solution gives a $n \times 1$ matrix.
+
+:::
+
+## Matrix Multiplication
+
+Matrices $A$ and $B$ can be multiplied together if $A$ is an $n \times p$ matrix and $B$ is an $p\times m$ matrix.
+The general element $c_{ij}$ of $n\times m$, $C=AB$, is found by pairing the $i^{th}$ row of $C$ with the $j^{th}$ column of $B$, and computing the sum of products of the paired terms.
+
+![Fig. 39](../media/25_4_Matrix_multiplication.png)
+
+### Details
+
+Matrices $A$ and $B$ can be multiplied together if $A$ is a $n\times p$ matrix and $B$ is a $p\times m$ matrix.
+Given the general element $c_{ij}$ of $n \times m$ matrix, $C=AB$ is found by pairing the $i^{th}$ row of $C$ with the $j^{th}$ column of $B$, and computing the sum of products of the paired terms.
+
+### Examples
+
+:::info Example: Matrices in R
+
+```text
+> A <- matrix(c(1,3,5,2,4,6),3,2)
+
+> A
+     [,1] [,2]
+[1,]    1    2
+[2,]    3    4
+[3,]    5    6
+
+> B <- matrix(c(1,1,2,3),2,2)
+
+> B
+     [,1] [,2]
+[1,]    1    2
+[2,]    1    3
+
+> A%*%B
+     [,1] [,2]
+[1,]    3    8
+[2,]    7   18
+[3,]   11   28
+```
+
+:::
+
+## More on Matrix Multiplication
+
+Let $A$, $B$, and $C$ be $m\times n$, $n\times l$, and $l\times p$ matrices, respectively.
+Then we have
+
+$$(AB)C=A(BC)$$
+
+In general, matrix multiplication is not commutative, that is $AB\neq BA$.
+
+We also have
+
+$$(AB)'=B'A'$$
+
+In particular, $(Av)'(Av)=v'A'Av$, when $v$ is a $n\times1$ column vector
+
+More obvious are the rules
+
+1. $A+(B+C)=(A+B)+C$
+
+1. $k(A+B)=kA+kB$
+
+1. $A(B+C)=AB+AC$
+
+where $k\in\mathbb{R}$ and when the dimensions of the matrices fit.
+
+## Linear Equations
+
+### Details
+
+General linear equations can be written in the form $Ax=b$.
+
+### Examples
+
+:::info Example
+
+The set of equations
+
+$$2x+3y=4$$
+
+$$3x+y=2$$
+
+can be written in matrix formulation as
+
+$$
+\begin{bmatrix}
+  2 & 3 \\
+  3 & 1
+\end{bmatrix}
+\begin{bmatrix}
+  x \\
+  y
+\end{bmatrix} =
+\begin{bmatrix}
+  4 \\
+  2
+\end{bmatrix}
+$$
+
+i.e. $A\underline{x} = \underline{b}$ for an appropriate choice of $A, \underline{x}$ and $\underline{b}$.
+
+:::
+
+## The Unit Matrix
+
+The $n\times n$ matrix
+
+$$
+I =
+  \left[
+    \begin{array}{cccc}
+      1 & 0 & \ldots & 0 \\
+      0 & 1 & 0 & \vdots \\
+      \vdots & 0 & \dots & 0 \\
+      0 & \ldots & 0 & 1
+    \end{array}
+  \right]
+$$
+
+is the identity matrix.
+This is because if a matrix $A$ is $n\times n$
+
+then $A I = A$ and $I A = A$
+
+## The Inverse of a Matrix
+
+If $A$ is an $n \times n$ matrix and $B$ is a matrix such that
+
+$$BA = AB = I$$
+
+then $B$ is said to be the inverse of $A$, written
+
+$$B = A ^{-1}$$
+
+Note that if $A$ is an $n \times n$ matrix for which an inverse exists, then the equation $Ax = b$ can be solved and the solution is $x = A^{-1} b$.
+
+### Examples
+
+:::info Example
+
+If matrix $A$ is:
+
+$$
+\begin{bmatrix}
+  2 & 3 \\
+  3 & 1
+\end{bmatrix}
+$$
+
+then $A ^{-1}$ is:
+
+$$
+\begin{bmatrix}
+  \displaystyle\frac{-1}{7} & \displaystyle\frac{3}{7} \\
+  \displaystyle\frac{3}{7} & \displaystyle\frac{-2}{7}
+\end{bmatrix}
+$$
+
+:::