Objectives
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Definitions of matrices and vectors.
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Learning some notation around matrices and vectors.
Section 2.1 Matrix and Vector Definition
Subsection 2.1.1 Matrices
Definition 2.1.1.
A matrix is a rectangular grid of numbers. An \(m\) by \(n\) matrix has \(m\) rows and \(n\) columns.
For example,
\begin{equation}
\begin{bmatrix}
1 \amp 2 \amp 11 \amp -1 \\ 3 \amp -2 \amp 3 \amp 4
\end{bmatrix}\tag{2.1.1}
\end{equation}
The size of a matrix is the number of rows and columns in the matrix. The number of rows is listed first. The size of the example above is 2 by 4.
The numbers in a matrix are called the entries or elements of the matrix. For example, for the matrix in (2.1.1), the entry on the first row and third column is 11. Often we will use the notation \(a_{1,3} = 11\text{,}\) where the subscript 1 is the row number and \(3\) is the column number.
Subsection 2.1.2 Vectors
Definition 2.1.2.
A vector is a matrix with one of its dimensions is 1. If a matrix only has one column it is called a column vector. If the matrix has only one row it is called a a row vector. The number of elements (numbers) in the vector is called the length.
As we investigate matrices, since vectors are in fact matrices, the properties will hold for vectors as well.
Subsection 2.1.3 Matrix, Vector and Scalar Notation
A scalar is a fancy term for a number. Mathematicians use this term to distinguish them from matrices and vectors, which are not scalars. Whenever variables are used for scalars, then lower case letters will be used. For example, \(a, n\) and \(x\) are scalar variables.
When we use variable names for matrices, we will use capital letters. For example, \(A, B\) and \(X\) are matrix (or vector) variables.
Often, vectors will be lower case with an arrow on top. For example, here are vectors: \(\vec{u}, \vec{v}, \vec{x}\text{.}\)
Remark 2.1.3.
In general, a matrix \(A\) can be written \([a_{ij}]\text{.}\) These will be used in proofs later on in this chapter and for defintions of operations. A vector \(\vec{v}\) can be written \([v_i]\) and although this notation can be used for both column and row vectors, unless denoted as such vectors will be consider as column vectors. We will see later in this chapter that we can simply write a row vector as a transformation of a column vector.
