Systems of linear equations — matrix form Ax = b

4.5.8 · Maths › Linear Algebra (Full)

$A x = b$ KYA hai?

$A x$ mein pack karne se equations kyun reproduce hoti hain? (Scratch se derive karo)

Product $A x$ ko define hi aise kiya gaya hai ki uski $i$ -th entry, $A$ ki row $i$ aur $x$ ka dot product hoti hai:

(A x)_{i} = j = 1 \sum n a_{ij} x_{j} .

Har $i$ ke liye $(A x)_{i} = b_{i}$ set karna literally equation $i$ likh deta hai. To $A x = b$ koi nayi rule nahi hai — yeh wohi $m$ equations hain, compressed form mein. Yahi wajah hai ki matrix multiplication exactly is tarah define hoti hai.

TEEN views (yahi gold hai)

Column view, derived

$A x$ ko columns ke hisaab se split karo. Maano $a_{(j)}$ , $A$ ka $j$ -th column hai. Tab

A x = [a_{(1)} a_{(2)} \dots a_{(n)}] x_{1} ⋮ x_{n} = x_{1} a_{(1)} + x_{2} a_{(2)} + \dots + x_{n} a_{(n)} .

Yeh step kyun? Har entry $(A x)_{i} = \sum_{j} a_{ij} x_{j}$ fixed $j$ ke liye term $x_{j} a_{ij}$ collect karti hai — jo exactly $x_{j}$ times column $a_{(j)}$ ki $i$ -th entry hai. $i$ par vector ke roop mein sum karne se $x_{j} a_{(j)}$ milta hai.

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

Solution kab exist karta hai / kab unique hota hai?

Rank kyun? Rank independent equations count karta hai. Agar $b$ equation block mein ek nayi independent "direction" add karta hai (rank badhti hai), to woh kuch aisa maang raha hai jo columns supply nahi kar sakte → inconsistent. Agar equations match karti hain lekin unknowns se kam hain, to bache hue unknowns free rehte hain.

Square case ( $m = n$ )

Worked examples

Common mistakes (steel-manned)

Active recall

Recall Forecast then verify

Padhne se pehle: ${x + y = 2 2 x + 2 y = 5$ ke solution type ka predict karo. Verify: rank $(A) = 1$ , rank $([A ∣ b]) = 2$ → no solution (inconsistent parallel lines).

Recall Feynman: ek 12-saal ke bacche ko explain karo

Ek recipe machine imagine karo. $A$ ke columns ingredients ke dabbe hain. $x$ batata hai ki har dabbe ke kitne scoops use karne hain. $b$ woh dish hai jo tum chahte ho. $A x = b$ solve karna matlab sahi number of scoops dhundhna hai taaki exactly woh dish bane. Kabhi ek perfect recipe hoti hai (unique), kabhi kaafi recipes kaam karti hain (infinite), aur kabhi dish ke liye ek aisa ingredient chahiye hota hai jo kisi bhi shelf par nahi hai (no solution).

A x

i

-th entry kya hoti hai?

A

ki row

i

aur

x

ka dot product, yaani

\sum_{j} a_{ij} x_{j}

Column view mein

A x

kya hota hai?

A

ke columns ki ek linear combination jisme weights

x_{j}

hote hain:

\sum_{j} x_{j} a_{(j)}

A x = b

kab solvable hota hai (existence condition)?

Exactly tab jab

b \in Col (A)

, yaani

rank (A) = rank ([A ∣ b])

UNIQUE solution ke liye Rouché–Capelli condition kya hai?

rank (A) = rank ([A ∣ b]) = n

(unknowns ki sankhya).

Invertible square

A

ke liye solution kya hai aur kyun?

x = A^{- 1} b

;

A x = b

A^{- 1}

se left-multiply karo to

I x = A^{- 1} b

milta hai.

A x = b

ka general solution structure kya hai?

x = x_{p} + x_{h}

: ek particular solution plus null space ke saare elements (

A x_{h} = 0

Ek system mein unknowns se zyada equations hone ke bawajood woh solvable kyun ho sakta hai?

Agar extra equations dependent (redundant) hain, to rank low rehti hai; solvability rank par depend karti hai, na ki row count par.

Jab

rank = r < n

ho to kitne free parameters hote hain?

n - r

free parameters.

Connections

Matrix multiplication — $A x$ define karta hai.
Gaussian elimination — $A x = b$ actually solve karne ka algorithm.
Rank of a matrix — existence aur uniqueness decide karta hai.
Column space and null space — solvability aur free parameters ke peeche ke do subspaces.
Determinant — square case mein invertibility test.
Inverse of a matrix — $x = A^{- 1} b$ deta hai.
Linear independence — unique solution $⟺$ columns independent hain.