Preface-3

Now we include 2 more columns in A The 4 columns are in 3-dimensional space

$A = \begin{bmatrix} 2 & 1 & 3 & 0 \\ 3 & 4 & 7 & 0 \\ 1 & 2 & 3 & -1 \end{bmatrix}$

Linear algebra aims for an understanding of every column space. Let me try this one.

Columns 1 and 2 produce the same plane as before (same $a_1$ and $a_2$)

Column 3 contributes nothing new because $a_3$ is on that plane: $a_3 = a_1 + a_2$

Column 4 is not on the plane: Adding in $c_4a_4$ raises or lowers the plane

The column space of this matrix $A$ is the whole 3-dimensional space: all points!

You see how we go a column at a time, left to right. Each column can be independent of the previous columns or it can be a combination of those columns. To produce every point in 3-dimensional space, you need three independent columns.

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Matrix Multiplication $A = CR$

Using the words "linear combination" and "independent columns" gives a good picture of that 3 by 4 matrix $A$. Column 3 is a linear combination: column 1 + column 2. Columns 1, 2, 4 are independent. The only way to produce the zero vector as a combination of the independent columns 1, 2, 4 is to multiply all those columns by zero.

We are so close to a key idea of Chapter 1 that IAaron to go on. Matrix multiplication isform is the perfect way to write down what we know. From the 4 columns of$A$ we pick out the independent columns$a_1, a_2, a_4$ in the column matrix$C$. Every column of$R$ tells us the combination of$a_1, a_2, a_4$ in$C$ that produces a column of$A$.$A$ equals$C$ times$R$:

$A = \begin{bmatrix} 2 & 1 & 3 & 0 \\ 3 & 4 & 7 & 0 \\ 1 & 2 & 3 & -1 \end{bmatrix} = \begin{bmatrix} 2 & 1 & 0 \\ 3 & 4 & 0 \\ 1 & 2 & -1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} = CR$

Column 3 of $A$ is dependent on columns 1 and 2 of $A$, and column 3 of $R$ shows how. Add the independent columns 1 and 2 of $C$ to get column $a_3 = a_1 + a_2 = (3,7,3)$ of$A$.

Matrix multiplication: Each column $j$ of $CR$ is $C$ times column $j$ of$R$.

Section 1.3 of the book will multiply a matrix times a vector (two ways). Then Section 1.4 will multiply a matrix times a matrix. This is the key operation of linear algebra. It is important that there is more than one good way to do this multiplication.

I am going to stop here. The normal purpose of the Preface is to tell you about the big picture. The next pages will give you two ways to organize this subject—especially the first seven chapters that more than fill up most linear algebra courses. Then come optional chapters, leading to the most active topic in applications today: deep learning.