Systems of linear equations — matrix form Ax = b

WHAT is $Ax=b$ ?

Definition System in matrix form

A system of $m$ equations in $n$ unknowns

\begin{aligned} a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n &= b_1\\ a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n &= b_2\\ &\;\;\vdots\\ a_{m1}x_1 + a_{m2}x_2 + \dots + a_{mn}x_n &= b_m \end{aligned}

is written compactly as $==Ax=b==$ , where

A=\begin{bmatrix}a_{11}&\cdots&a_{1n}\\ \vdots & & \vdots\\ a_{m1}&\cdots&a_{mn}\end{bmatrix}\in\mathbb{R}^{m\times n},\quad x=\begin{bmatrix}x_1\\\vdots\\x_n\end{bmatrix},\quad b=\begin{bmatrix}b_1\\\vdots\\b_m\end{bmatrix}.

WHY does packing into $Ax$ reproduce the equations? (Derive from scratch)

The product $Ax$ is defined so that its $i$ -th entry is the dot product of row $i$ of $A$ with $x$ :

(Ax)_i \;=\; \sum_{j=1}^{n} a_{ij}\,x_j.

Setting $(Ax)_i = b_i$ for each $i$ literally writes out equation $i$ . So $Ax=b$ is not a new rule — it is the same $m$ equations, compressed. That's the whole reason matrix multiplication is defined the way it is.

The THREE views (this is the gold)

Column view, derived

Split $Ax$ by columns. Let $a_{(j)}$ be the $j$ -th column of $A$ . Then

Ax=\begin{bmatrix} a_{(1)} & a_{(2)} & \cdots & a_{(n)}\end{bmatrix}\begin{bmatrix}x_1\\\vdots\\x_n\end{bmatrix} = x_1 a_{(1)} + x_2 a_{(2)} + \dots + x_n a_{(n)}.

Why this step? Each entry $(Ax)_i=\sum_j a_{ij}x_j$ collects, for fixed $j$ , the term $x_j a_{ij}$ — which is exactly $x_j$ times the $i$ -th entry of column $a_{(j)}$ . Summing over $i$ as a vector gives $x_j a_{(j)}$ .

Figure — Systems of linear equations — matrix form Ax = b

WHEN does a solution exist / is it unique?

WHY rank? Rank counts independent equations. If $b$ adds a new independent "direction" to the equation block (rank goes up), it's demanding something the columns can't supply → inconsistent. If equations match but there are fewer of them than unknowns, the leftover unknowns float free.

The square case ( $m=n$ )

Worked examples

Worked example Unique solution (square, invertible)

Solve $\begin{cases} x+2y=5\\ 3x+4y=11\end{cases}$ . $A=\begin{bmatrix}1&2\\3&4\end{bmatrix},\ b=\begin{bmatrix}5\\11\end{bmatrix}$ . Step 1: $\det A = 1\cdot4-2\cdot3 = -2 \neq 0.$ Why? Nonzero det ⇒ unique solution exists. Step 2: $A^{-1}=\frac{1}{-2}\begin{bmatrix}4&-2\\-3&1\end{bmatrix}$ . Why? $2\times2$ inverse swaps diagonal, negates off-diagonal, divides by det. Step 3: $x=A^{-1}b=\frac{1}{-2}\begin{bmatrix}4\cdot5-2\cdot11\\-3\cdot5+1\cdot11\end{bmatrix}=\frac{1}{-2}\begin{bmatrix}-2\\-4\end{bmatrix}=\begin{bmatrix}1\\2\end{bmatrix}.$ Check (column view): $1\begin{bmatrix}1\\3\end{bmatrix}+2\begin{bmatrix}2\\4\end{bmatrix}=\begin{bmatrix}5\\11\end{bmatrix}=b.$ ✓

Common mistakes (steel-manned)

Active recall

Recall Forecast then verify

Before reading: predict the solution type of $\begin{cases}x+y=2\\ 2x+2y=5\end{cases}$ . Verify: rank $(A)=1$ , rank $([A|b])=2$ → no solution (inconsistent parallel lines).

Recall Feynman: explain to a 12-year-old

Imagine a recipe machine. The columns of $A$ are jars of ingredients. $x$ tells you how many scoops of each jar to use. $b$ is the dish you want. Solving $Ax=b$ means finding the right number of scoops to make exactly that dish. Sometimes there's one perfect recipe (unique), sometimes many recipes work (infinite), and sometimes the dish needs an ingredient you just don't have on any shelf (no solution).

What does the

i

-th entry of

Ax

equal?

The dot product of row

i

A

with

x

, i.e.

\sum_j a_{ij}x_j

In the column view, what is

Ax

A linear combination of the columns of

A

with weights

x_j

\sum_j x_j a_{(j)}

When is

Ax=b

solvable (existence condition)?

Exactly when

b\in\operatorname{Col}(A)

, i.e.

\operatorname{rank}(A)=\operatorname{rank}([A|b])

State the Rouché–Capelli condition for a UNIQUE solution.

\operatorname{rank}(A)=\operatorname{rank}([A|b])=n

(number of unknowns).

For invertible square

A

, what is the solution and why?

x=A^{-1}b

; left-multiply

Ax=b

A^{-1}

to get

Ix=A^{-1}b

General solution structure of

Ax=b

x=x_p+x_h

: one particular solution plus all of the null space (

Ax_h=0

Why can a system with more equations than unknowns still be solvable?

If the extra equations are dependent (redundant), rank stays low; solvability depends on rank, not row count.

How many free parameters when

\operatorname{rank}=r<n

n-r

free parameters.

Connections

Matrix multiplication — defines $Ax$ .
Gaussian elimination — the algorithm to actually solve $Ax=b$ .
Rank of a matrix — decides existence & uniqueness.
Column space and null space — the two subspaces behind solvability and free parameters.
Determinant — square-case invertibility test.
Inverse of a matrix — gives $x=A^{-1}b$ .
Linear independence — unique solution $\iff$ columns independent.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, $Ax=b$ ka matlab koi naya jaadu nahi hai — ye sirf bahut saari linear equations ko ek clean form me likhne ka tareeka hai. $A$ me coefficients aate hain, $x$ me unknowns, aur $b$ me right-hand side. Jab tum $Ax$ ka $i$ -th element nikaalte ho, wo $A$ ke row $i$ aur $x$ ka dot product hota hai — yaani wahi original equation. Isliye matrix multiplication aise hi define kiya gaya hai, taaki ye trick automatically kaam kare.

Sabse important baat: ek hi equation ke do views hain. Row view me har equation ek line (ya hyperplane) hai, aur solution unka intersection hai. Column view me $Ax$ matlab columns ka linear combination hai — $x_j$ batata hai ki har column ko kitne "scoops" me milana hai taaki $b$ ban jaaye. Isliye solution tabhi milega jab $b$ , columns ke span (column space) ke andar ho. Ye column view 80/20 idea hai — isse existence ekdum clear ho jaata hai.

Solution kitne honge, ye rank decide karta hai. Agar $\operatorname{rank}(A) = \operatorname{rank}([A|b]) = n$ (unknowns), to ek unique solution. Agar dono rank equal hain par $n$ se kam, to infinite solutions ( $n-r$ free variables). Aur agar augmented matrix ka rank zyada ho gaya, matlab system inconsistent hai — koi solution nahi. Square aur invertible $A$ ke liye shortcut: $x=A^{-1}b$ , lekin yaad rakho — matrix me "divide by $A$ " jaisa kuch nahi hota, sirf inverse se multiply karo, aur order $A^{-1}b$ hi rahe.

Exam me sabse common galti: $\det$ ko har system pe lagaa dena. $\det$ sirf square matrix ke liye hai; non-square ke liye rank use karo. "RACE" mnemonic yaad rakho aur tum classify karne me kabhi nahi phasoge.

Systems of linear equations — matrix form Ax = b

WHAT is $Ax=b$ ?

WHY does packing into $Ax$ reproduce the equations? (Derive from scratch)

The THREE views (this is the gold)

Column view, derived

WHEN does a solution exist / is it unique?

The square case ( $m=n$ )

Worked examples

Common mistakes (steel-manned)

Active recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Linear Algebra (Full)

Connections