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Vectors

A row vector (also called a row matrix) is a $1$-by-$n$ matrix; a column vector (also called a column matrix) is a $n$-by-$1$ matrix. Both kinds of vectors get their names from their resemblance to the rows and columns of a matrix, respectively.

Examples

A vector can be represented as a column matrix, which allows for multiplying a matrix by a vector.

Properties

One way to represent column vectors is as transposes of row vectors.

The product of a $1$-by-$n$ row vector and a $n$-by-$1$ column vector is a $1$-by-$1$ value; this is where the dot product comes from!

The product of a $n$-by-$1$ column vector and a $1$-by-$n$ row vector is a $n$-by-$n$ matrix.

(Video) What is a Column Vector?

by numericalmethodsguy

Here's a nice short video by numericalmethodsguy explaining what a column vector is.

(Video) What is a Row Vector?

by numericalmethodsguy

Here's a short video by numericalmethodsguy explaining what a row vector is.

Row and Column Matrices

In this lesson, we will talk about two specific types of matrices: row and column matrices.

A row matrix is a matrix with only one row, of the size $1$-by-$n$. Here is an example of a row matrix:

$$\begin{bmatrix} 1&3&6&8 \\ \end{bmatrix}$$

The number of rows is $1$. The elements of the matrix are arranged in just one row.

A column matrix is a matrix with just column, of the size $n$-by-$1$. Here is an example of a column matrix:

$$\begin{bmatrix} 1 \\ 5 \\ 8 \\ 3 \\ \end{bmatrix}$$

The number of columns is $1$. The elements are arranged in only one column.

The shape of the matrix is rectangular. This is not a square matrix.