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4.5.11Linear Algebra (Full)

Pivot positions, free variables

1,605 words7 min readdifficulty · medium
🇮🇳 Yeh note Hinglish mein padho →

WHAT are pivots and free variables?

WHY this distinction exists: After row reduction, each equation that still "lives" lets you solve for exactly one leading variable in terms of the others. Columns that never earn a leading 1 are never solved for — they stay as parameters.

HOW to find them (the procedure)

  1. Row-reduce AA (or augmented [A∣b][A\mid \mathbf b]) to echelon form.
  2. The first nonzero entry of each nonzero row marks a pivot position → its column is a pivot column.
  3. Pivot columns ⇒ pivot variables. Non-pivot columns ⇒ free variables.
  4. Count: if AA is m×nm\times n with rr pivots, then #free variables=n−r.\#\text{free variables} = n - r.
Figure — Pivot positions, free variables

WHY it controls the solution set

Worked Example 1 — one free variable (a line)

Solve [121243]x=[13].\begin{bmatrix} 1 & 2 & 1\\ 2 & 4 & 3 \end{bmatrix}\mathbf x = \begin{bmatrix}1\\3\end{bmatrix}.

Step 1. R2←R2−2R1R_2 \leftarrow R_2 - 2R_1: [12110011].\left[\begin{array}{ccc|c} 1 & 2 & 1 & 1\\ 0 & 0 & 1 & 1\end{array}\right]. Why this step? We zero out below the first pivot to expose echelon structure.

Step 2. Pivots in columns 1 and 3 → x1,x3x_1,x_3 pivot variables; column 2 has no pivot → x2x_2 is free. Why? Column 2 never gets a leading entry.

Step 3. Back-substitute. Row 2: x3=1x_3 = 1. Row 1: x1=1−2x2−x3=−2x2x_1 = 1 - 2x_2 - x_3 = -2x_2. Why? Solve each pivot variable in terms of the free one.

Result: with x2=tx_2=t, x=[001]+t[−210].\mathbf x = \begin{bmatrix}0\\0\\1\end{bmatrix} + t\begin{bmatrix}-2\\1\\0\end{bmatrix}. n−r=3−2=1n-r = 3-2 = 1 free variable ⇒ a line. ✓

Worked Example 2 — no free variables (unique solution)

[1102]x=[34].\begin{bmatrix}1 & 1\\ 0 & 2\end{bmatrix}\mathbf x=\begin{bmatrix}3\\4\end{bmatrix}.

Step 1. R2←12R2R_2 \leftarrow \tfrac12 R_2: x2=2x_2 = 2. Why? Make the leading entry 1. Step 2. Both columns are pivot columns → no free variables. n−r=2−2=0n-r = 2-2 = 0. Step 3. x1=3−x2=1x_1 = 3 - x_2 = 1. Unique solution (1,2)(1,2). ✓

Worked Example 3 — inconsistent (free count irrelevant)

[1−12005].\left[\begin{array}{cc|c}1 & -1 & 2\\ 0 & 0 & 5\end{array}\right]. Row 2 reads 0=50 = 5. The augmented column is a pivot column ⇒ no solution. Why this matters: free variables describe a solution set's dimension only when a solution exists. Always check consistency first.

Common Mistakes (Steel-manned)

Recall Feynman: explain to a 12-year-old

Imagine a quiz where each rule lets you figure out one hidden number. If you have 3 hidden numbers and only 2 good rules, one number is never pinned down — you get to pick it yourself, and the other two then follow. That "pick-it-yourself" number is a free variable; the ones the rules nail down are pivot variables. The more free numbers you can pick, the bigger the family of answers (one pick = a line, two picks = a plane).

Flashcards

What is a pivot position?
A location in AA holding a leading entry of its RREF; its column is a pivot column.
How do you get the number of free variables?
n−rn - r, where nn = number of columns/variables and rr = rank (number of pivots).
Pivot variable vs free variable?
Pivot variable = variable in a pivot column (solved by an equation); free variable = variable in a non-pivot column (assignable freely).
If a system has free variables, must it have infinitely many solutions?
No — only if it is consistent; an inconsistent system has none regardless.
What does "augmented column is a pivot column" mean?
A row like [0 ⋯ 0∣c][0\,\cdots\,0\mid c], c≠0c\neq0 → the system is inconsistent (no solution).
Geometrically, what does kk free variables give?
A kk-dimensional solution flat (point if 0, line if 1, plane if 2).
Are pivot positions unique to a matrix?
Yes — RREF is unique, so pivot positions are determined by AA.
Why isn't "# free variables = # zero rows" correct?
Free variables count non-pivot columns (n−rn-r), unrelated to zero rows.

Connections

Concept Map

reveals

located in

gives

gives

counted as

used in

number equals

k free means

b column pivot means

Row reduce to RREF

Pivot position leading 1

Pivot column

Non-pivot column

Pivot variable

Free variable

rank r = number of pivots

free = n - r

Solution set shape

Consistency check

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, jab tum koi linear system Ax=bA\mathbf{x}=\mathbf{b} solve karte ho, to row reduction ke baad har row ka jo pehla nonzero leading entry hota hai usse hum pivot bolte hain. Jis column me pivot aaya, wo pivot column — aur uska variable "pin" ho gaya, matlab equation usse solve kar degi. Jis column me koi pivot nahi aaya, uska variable free variable hai — tum usse koi bhi value de sakte ho, apni marzi se.

Sabse important formula simple hai: free variables = n−rn - r, jahan nn total variables (columns) hain aur rr = rank = number of pivots. Yaad rakho — free variables zero rows se nahi aate, balki non-pivot columns se aate hain. Ye chhoti si galti bahut log karte hain.

Geometry me iska matlab: agar 0 free variables hue to ek hi unique solution (ek point). 1 free variable = ek line. 2 free variables = ek plane. Lekin ek condition: ye sab tabhi valid hai jab system consistent ho. Agar augmented column khud ek pivot column ban jaye (jaise row 0=50=5), to solution hi nahi hai — chahe kitne bhi free variables dikh rahe hon.

Toh strategy: pehle RREF nikalo, pivots count karo, consistency check karo, phir n−rn-r se free variables count karke solution set ka shape bata do. Bas yahi 80/20 hai is topic ka.

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