# Matrix notation

### Matrix Dimensions

Whenever we say "A is an m by n matrix," or simply "A is m x n," for some positive integers m and n, this means that A has m rows and n columns. An example of a 3 x 5 matrix is:

### Vectors

A vector can be seen as either a 1 x n matrix in the case of a row vector, or an n x 1 matrix in the case of a column vector. Below, v is a 3 x 1 column vector and w is a 1 x 4 row vector.

Column vectors are much more commonly used than row vectors.

### Linear combinations

A linear combination of vectors, v_{1}, ..., v_{n} is any vector of the form

x_{1}v_{1} + ... + x_{n}v_{n}

for some numbers x_{1}, ..., x_{n}. For example, the vector,

is a linear combination of:

because

or

v = 3v_{1} + 5v_{2} - 3v_{3}

### ℝ^{n} and standard basis vectors

If n is any positive integer, an n-dimensional vector with real entries is called an n-vector, and the set of all n-vectors is called n-dimensional space, n-space, or ℝ^{n}. For example, ℝ^{3} consists of all 3-vectors of the form

where a, b, and c are real numbers. In ℝ^{n}, we define e_{i} for all i = 1, 2, ... n as the column vector with a 1 in the ith position (counting from the top-down) and 0 everywhere else. For example, in ℝ^{3},

The e_{i}'s are called the standard basis vectors of ℝ^{n} because any n-vector in ℝ^{n} can be represented as a linear combination of the standard basis vectors. For example, in ℝ^{3}, the vector

can be represented as

v = ae_{1} + be_{2} + ce_{3}

### Matrices as collections of rows and columns

Often, it is useful to think about an m × n matrix A as a collection of its n column vectors, each of which has m dimensions. For example, the 3 × 4 matrix,

can be viewed as the following set of 4 vectors, each of 3 dimensions.

Here, the curly braces, {}, are being used to denote a set of objects. If the column vectors of A are v_{1}, ..., v_{n} in that order, then we can represent A as,

which we write as A = [v_{1} v_{2} ... v_{n}]. Similarly, we can also view A as the collection of its m row vectors, each of which has n dimensions. If the row vectors of A are r_{1}, ..., r_{m} in that order, then we can represent A as

### Entries of a matrix

Each number inside a matrix is called an entry. To refer to a specific entry of a matrix we use the i,j^{th} notation: for some positive integers i and j, the i,j^{th} entry of a matrix A, denoted a_{i,j} is the entry in the i^{th} row of the j^{th} column. We always count rows from top to bottom and columns from left to right.

In the matrix above, if i = 2 and j = 4, the i,j^{th} entry would be -3.