Square Matrices and Column Vectors

Although everything above has been stated in terms of general rectangular matrices, for the rest of this tutorial, we'll consider only two kinds of matrices (but of any dimension): square matrices, where the number of rows is equal to the number of columns, and column matrices, where there is only one column. These column matrices are often called ``vectors'', and there are many applications where they correspond exactly to what you commonly use as sets of coordinates for points in space. In the two-dimensional $ x$-$ y$ plane, the coordinates $ (1, 3)$ represent a point that is one unit to the right of the origin (in the direction of the $ x$-axis), and three units above the origin (in the direction of the $ y$-axis). That same point can be written as the following column vector:

$\displaystyle \begin{pmatrix}1 \\ 3 \end{pmatrix}. $

If you wish to work in three dimensions, you'll need three coordinates to locate a point relative to the (three-dimensional) origin--an $ x$-coordinate, a $ y$-coordinate, and a $ z$-coordinate. So the point you'd normally write as $ (x, y, z)$ can be represented by the column vector:

$\displaystyle \begin{pmatrix}x \\ y \\ z \end{pmatrix}. $

Quite often we will work with a combination of square matrices and column matrices, and in that case, if the square matrix has dimensions $ n\times n$, the column vectors will have dimension $ n\times 1$ ($ n$ rows and 1 column)1.

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