Preface-2

That picture illustrated two basic operations—adding vectors ($a_1 + a_2$) and multiplying a vector by 2. Combining those operations produced a "linear combination" ($2a_1 + a_2$):

$$\text{Linear combination} = c a_1 + d a_2 \quad \text{for any numbers } c \text{ and } d$$

Those numbers ($c$) and ($d$) can be negative. In that case ($c a_1$) and ($d a_2$) will reverse their directions: they go right to left. Also very important, ($c$) and ($d$) can involve fractions. Here is a picture with a lot more linear combinations. Eventually we want all vectors ($c a_1 + d a_2$).

Here is the key! The combinations ($c a_1 + d a_2$) fill a whole plane. It is an infinite plane in 3-dimensional space. By using more and more fractions and decimals (c) and (d), we fill in a complete plane. Every point on the plane is a combination of ($a_1$) and ($a_2$).

Now comes a fundamental idea in linear algebra: a matrix. The matrix ($A$) holds ($n$) column vectors ($a_1, a_2, \dots, a_n$). At this point our matrix has two columns ($a_1$) and ($a_2$), and those are vectors in 3-dimensional space. So the matrix has three rows and two columns.

$$\begin{aligned} &3 \text{ by } 2 \text{ matrix} \\ &m = 3 \text{ rows} \\ &n = 2 \text{ columns} \end{aligned} \quad A = \begin{bmatrix} a_1 & a_2 \end{bmatrix} = \begin{bmatrix} 2 & 1 \\ 3 & 4 \\ 1 & 2 \end{bmatrix}$$

The combinations of those two columns produced a plane in three-dimensional space. There is a natural name for that plane. It is the column space of the matrix. For any ($A$), the column space of ($A$) contains all combinations of the columns.

Here are the four ideas introduced so far. You will see them all in Chapter 1.

1. Column vectors ($a_1$) and ($a_2$) in three dimensions
2. Linear combinations ($c a_1 + d a_2$) of those vectors
3. The matrix ($A$) contains the columns ($a_1$) and ($a_2$)
4. Column space of the matrix = all linear combinations of the columns = plane