Linear Algebra/Matrices and Vectors/

Matrices

A matrix is an array of numbers arranged into rows and columns. Some examples of matrices are,

${\displaystyle A={\begin{bmatrix}2&4&6\\0&-5&7.105\\1&-3&2\end{bmatrix}},\quad B={\begin{bmatrix}-3&-4\\\pi &{\sqrt {2}}\end{bmatrix}},{\mbox{ and }}\,\,\,C={\begin{bmatrix}-3&-5&0&0\\-1&4.56&3.28&19\end{bmatrix}}.}$

When describing matrices we indicate the number of rows first, then the number of columns. For example, the matrix ${\displaystyle C}$  with two rows and four columns is said to be a ${\displaystyle 2\times 4}$  matrix.

It is standard notation to name matrices with capital letters and to use lower case letters with subscripts to identify particular entries in a matrix.

For example, to identify the entry in row 1 and column 3 of matrix ${\displaystyle A}$  we would write ${\displaystyle a_{13}}$ . To indicate that this entry is a six we would write the equation ${\displaystyle a_{13}=6}$ .

Two matrices are considered to be equal only if they are the same size and every pair of corresponding elements are equal.

A column matrix is a matrix with only one column. Similarly, a row matrix has only one row.

Vectors

A vector is an object often defined by a long list of properties. However, for now we will avoid the more complicated definition, and just say that a vector is an ordered list of numbers. Later we will see that vectors can really be much more.

An ordered pair, ${\displaystyle (x,y)}$ , that is used to identify a point in the plane can be considered to be a vector.

Similarly, an ordered triple, ${\displaystyle (x,y,z)}$  is a vector.

Obviously, row and column matrices can also be considered to be a vector.

It is common to name vectors using variables with arrows above.

For example, we might write ${\displaystyle {\vec {v}}=(2,3,5,-4),{\mbox{ or }}{\vec {w}}={\begin{bmatrix}4\\5\\0\end{bmatrix}}}$ .

For the most part, it will convenient to think of vectors as column matrices.