Foundations — Pivot positions, free variables

Before we can talk about "pivots" and "free variables", we must earn every piece of notation the parent note throws at you. We build them one at a time, each on top of the last. Nothing is assumed.

1. The unknowns $x_1,x_2,\dots ,x_n$ — and what a subscript means

Why not $x,y,z$? Once you have more than three unknowns you run out of letters. Writing $x_1,\dots ,x_n$ lets us talk about however many unknowns at once, and the letter $n$ stands for "the total number of unknowns."

2. What is a linear equation? (the atom)

Now that $x_1,x_2,\dots$ mean "labelled unknowns," we can build the smallest object of the whole subject.

The picture — read the figure. The figure below plots the equation $2x_1+3x_2=7$. The horizontal axis is $x_1$, the vertical axis is $x_2$. The blue line is the set of all pairs $(x_1,x_2)$ that make the equation true; the orange dots are two example solutions sitting on it. Notice the line is perfectly straight — that straightness is exactly what "linear" buys us.

Why the topic needs it. The entire subject is about systems of these — several straight constraints at once. If the equations bent or curved, none of the tidy counting rules would hold. Straightness is what makes everything downstream (pivots, ranks, dimensions) work.

3. The letter $n$ — counting the unknowns (columns)

The picture. Line up your unknowns left to right. There are $n$ slots. Later, each slot becomes a column of numbers — so "$n$ columns" and "$n$ variables" will turn out to be the same count. Hold on to that fact: it becomes the heart of the free-variable formula once we have defined the last symbol it needs.

4. Stacking equations into a system and reading it as a grid

The idea that unlocks everything: the unknowns just mark positions. We can strip the $x$'s away and keep only the numbers, lined up in a grid:

$\begin{array}{cccc}1& 2& 1& 1\\ 2& 4& 3& 3\end{array}$

Each row is one equation. Each column on the left holds the numbers multiplying one particular unknown. That grid of numbers is a matrix.

5. What is a matrix? Rows, columns, and the size $m\times n$

Read the figure. The figure shows our $2\times 3$ grid of numbers. The orange arrow on the left spans the two rows and is labelled "$m=2$ rows (equations)". The green arrow on top spans the three columns and is labelled "$n=3$ columns (variables)". Fix these two directions in your mind: rows run across, columns run down.

Why the topic needs it. Every rule in the parent note is phrased as "if $A$ is $m\times n$ with $r$ pivots…". You cannot use the free-variable formula if you can't tell which count is $m$ and which is $n$. $n$ is the one that matters for free variables because free variables come from columns.

6. The compact notation $A\mathbf{x}=\mathbf{b}$

The bold in $\mathbf{x}$ and $\mathbf{b}$ signals "this is a whole column of numbers", not a single number. The plain $A\mathbf{x}=\mathbf{b}$ is just a shorthand for the entire system at once.

7. The augmented matrix $[A\mid \mathbf{b}]$

Why the topic needs it. The parent note's consistency test — "is the augmented column a pivot column?" — is a statement about this glued-on last column. You can only ask that question if you've kept $\mathbf{b}$ attached.

8. Row reduction and echelon form (borrowed tool — why we use it)

The three legal moves (see Row Reduction and Echelon Forms):

1. Swap two rows.
2. Multiply a row by a nonzero number.
3. Add a multiple of one row to another.

Read the figure. The figure shows a reduced grid. The red bold numbers are the leading entries — the first nonzero number of each nonzero row — and the blue staircase line traces how each one steps down-and-right from the last. Column 2 (green note) never hosts a leading number; keep an eye on it, it becomes our first free column.

9. Leading entry, pivot position, pivot column — the payoff terms

Now every symbol in the parent's core definitions is earned.

The picture. In the staircase of Figure 3, a pivot is the "corner" number where each step begins. Its column is committed to solving one particular unknown; a non-pivot column is a column no step ever landed on.

10. Rank $r$ — just "how many pivots"

Why the topic needs it. Because "number of pinned-down unknowns" $=$ "number of pivots" $=r$, and "leftover free unknowns" $=n-r$. The whole counting formula lives or dies on knowing $r$ is the pivot count.

11. Consistency — is there any solution at all?

Why it comes before counting. Free variables measure the size of the solution set — but only if a solution exists. This is the separate check the parent note keeps warning about, expanded in Existence and Uniqueness of Solutions. (Recall from Section 6 that a homogeneous system $A\mathbf{x}=0$ is never in this trap — it is always consistent.)

12. Putting the count to work — what $n-r$ means

Here is the payoff that ties every symbol together. Assume the system is consistent (otherwise there are zero solutions, full stop). Then:

So the logic chain is: consistent? → if no, zero solutions; if yes, look at $n-r$ → $0$ means unique, $>0$ means infinitely many. That single number decides the entire qualitative outcome.

13. The bracket-vector solution form (the $\left[\begin{array}{c}\cdots \end{array}\right]+t\left[\begin{array}{c}\cdots \end{array}\right]$ notation)

The parent's answer to Example 1 was $\mathbf{x}=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]+t\left[\begin{array}{c}-2\\ 1\\ 0\end{array}\right].$

Read the figure. The 3-D figure plots this solution. The orange dot is the fixed start point $(0,0,1)$. The green arrow is the direction $(-2,1,0)$ you slide along as you turn the knob $t$. The blue line is the full set of solutions — every point on it is a valid $\mathbf{x}$ for some value of $t$. (Deeper in Null Space and Solution Sets of Ax=0.)

Prerequisite map

The diagram below is a flow chart: each box is one concept from this page, and each arrow means "you need the box at the tail before the box at the head makes sense." Read it top to bottom — start at "Linear equation" and follow the arrows; every path eventually funnels into the parent topic at the bottom. It is just a visual table of contents for the dependency order we built above.

Equipment checklist

Reveal each line and check you can answer without hesitation.

Connections

• Pivot positions, free variables
• Row Reduction and Echelon Forms
• Rank and the Rank-Nullity Theorem
• Null Space and Solution Sets of Ax=0
• Existence and Uniqueness of Solutions
• Linear Independence
• Basis and Dimension