Subspaces — four fundamental subspaces of a matrix

4.5.44 · Maths › Linear Algebra (Full)

1. Subspace kya hota hai? (pehle foundation)

YE RULES KYUN? Ek subspace ek "origin se guzarne wali flat cheez" honi chahiye (ek line, plane, hyperplane...). Agar tum scale aur add karo aur kabhi bahar na jao, toh tumhare paas ek poora vector world hai bina kisi edge ke. Ek line jo origin se nahi guzarti wo rule 1 fail karti hai, isliye wo subspace nahi hai.

2. Charon subspaces scratch se banana

Maano $A$ ek $m \times n$ matrix hai. Iske columns aur rows likhte hain: $A = ∣ a_{1} ∣ \dots ∣ a_{n} ∣, A \in R^{m \times n} .$

(a) Column space $C (A)$ — reachable outputs

KYUN: $A x = x_{1} a_{1} + \dots + x_{n} a_{n}$ literally columns ka combination hai. Toh saare possible outputs $b = A x$ ka set exactly columns ka span hai. KAISE use karein: $A x = b$ solvable hai $⟺ b \in C (A)$ .

(b) Null space $N (A)$ — inputs jo zero ho jaate hain

KYUN ye subspace hai: agar $A x = 0$ aur $A y = 0$ toh $A (c x + d y) = c A x + d A y = 0$ . Closed. MATLAB: ye woh inputs hain jinhein machine bilkul destroy kar deti hai.

(c) Row space $C (A^{T})$ — aur (d) Left null space $N (A^{T})$

Baaki do sirf transpose ke column space aur null space hain: $C (A^{T}) \subseteq R^{n} (rows ka span), N (A^{T}) \subseteq R^{m} ({y : A^{T} y = 0}) .$

Figure — Subspaces — four fundamental subspaces of a matrix

3. Dimensions: rank theorem (derive karo)

Maano $r = rank (A) =$ row reduction ke baad pivots ki sankhya.

Derivation. $A$ ko RREF mein row reduce karo.

Pivot columns linearly independent hote hain aur $C (A)$ ko span karte hain $\Rightarrow dim C (A) = r$ .
Row operations row space nahi badlate, aur RREF ki $r$ nonzero rows independent hain $\Rightarrow dim C (A^{T}) = r$ . Toh row rank = column rank. Isliye "rank" ek hi number hota hai.
Free variables = $n - r$ . Har free variable $A x = 0$ ka ek independent special solution deta hai $\Rightarrow dim N (A) = n - r$ .
$A^{T}$ par usi argument se (size $n \times m$ , rank bhi $r$ hai): $dim N (A^{T}) = m - r$ .

4. Orthogonality: gehra pairing

$N (A) ⊥ C (A^{T})$ ka derivation. Agar $x \in N (A)$ toh $A x = 0$ . $A$ ki $i$ -wein row $r_{i}$ hai, aur $(A x)_{i} = r_{i} \cdot x = 0$ . Toh $x$ har row ke orthogonal hai, isliye unke poore span $C (A^{T})$ ke bhi. Kyunki $dim N (A) + dim C (A^{T}) = n$ aur dono sirf $0$ share karte hain, milke ye $R^{n}$ bharte hain. ∎

5. Worked examples

Worked example

$A = [122436]$ ke charon subspaces nikalo Step 1 — Row reduce. $R_{2} \to R_{2} - 2 R_{1}$ : $[102030]$ . Kyun? RREF pivots expose karta hai → rank. Yahan $r = 1$ ( $m = 2, n = 3$ ). Step 2 — Column space. Pivot column 1 hai, toh $C (A) = span {(1, 2)}$ , $R^{2}$ mein ek line. Kyun? Pivot columns column space span karte hain; column 2,3 column 1 ke multiples hain. Step 3 — Row space. Nonzero RREF row: $C (A^{T}) = span {(1, 2, 3)}$ , $R^{3}$ mein ek line. Step 4 — Null space. $x_{1} + 2 x_{2} + 3 x_{3} = 0$ . Free: $x_{2}, x_{3}$ . Special solutions: set $x_{2} = 1, x_{3} = 0 \Rightarrow (- 2, 1, 0)$ ; set $x_{2} = 0, x_{3} = 1 \Rightarrow (- 3, 0, 1)$ . Toh $N (A) = span {(- 2, 1, 0), (- 3, 0, 1)}$ , dim $= n - r = 2$ . Kyun? 3 unknowns mein ek equation 2 free directions chodti hai. Step 5 — Left null space. $A^{T} y = 0$ solve karo, yaani $y_{1} (1, 2, 3) + y_{2} (2, 4, 6) = 0$ . Row 2 $= 2 \times$ row 1 hai, isliye $y_{1} + 2 y_{2} = 0 \Rightarrow y = (- 2, 1)$ . $N (A^{T}) = span {(- 2, 1)}$ , dim $= m - r = 1$ . Orthogonality check: $(1, 2) \cdot (- 2, 1) = 0$ ✓ (column $⊥$ left-null).

6. Forecast-then-Verify drill

Reduce karne se pehle, charon dimensions predict karo, phir check karo. $A$ ek $4 \times 6$ matrix hai rank $3$ ke saath. Forecast: $dim C (A) = 3$ , $dim C (A^{T}) = 3$ , $dim N (A) = 6 - 3 = 3$ , $dim N (A^{T}) = 4 - 3 = 1$ . Sums: $3 + 3 = 6 = n$ ✓, $3 + 1 = 4 = m$ ✓. Agar tumhara forecast sahi sum nahi karta, toh tumhara rank guess galat hai.

Recall Feynman: 12-saal ke bacche ko samjhao

Socho ek fancy juice mixer (matrix) hai. Tum usme fruits daalo (input vectors).

Kuch fruit combos kuch nahi dete — mixer unhe cancel kar deta hai. Un "waste" recipes ka collection null space hai.
Jo bhi alag-alag juices tum possibly bana sakte ho, woh column space hai.
Mixer sirf itne hi distinct juices bana sakta hai — woh count rank hai.
Aur kuch juices hain jo tum kabhi nahi bana sakte chahe kuch bhi karo — woh "impossible juice" zone left null space hai. Inme se do tumhare recipes (inputs) ke baare mein baat karte hain, do juices (outputs) ke baare mein, aur jo recipes bacha deti hain woh hamesha important recipes ke bilkul right angles par hoti hain.

Connections

Rank of a Matrix — woh ek number $r$ jo charon dimensions control karta hai.
Rank–Nullity Theorem — $dim C (A^{T}) + dim N (A) = n$ .
Orthogonal Complements — kyun ye pairs perpendicular hain.
Solving Ax=b — solvability ( $b \in C (A)$ ) aur general solution (particular $+ N (A)$ ).
RREF and Pivots — computational engine.
Least Squares & Projections — $C (A)$ aur $N (A^{T})$ directly use karta hai.

Flashcards

m \times n

matrix

A

ke chaar fundamental subspaces kya hain?

Column space

C (A)

, Row space

C (A^{T})

, Null space

N (A)

, Left null space

N (A^{T})

Kaun se subspaces input space

R^{n}

mein rehte hain?

Row space

C (A^{T})

aur Null space

N (A)

Kaun se subspaces output space

R^{m}

mein rehte hain?

Column space

C (A)

aur Left null space

N (A^{T})

dim C (A) = ?

r

(the rank).

dim N (A) = ?

n - r

dim N (A^{T}) = ?

m - r

Row rank = column rank kyun hota hai?

RREF

r

independent nonzero rows aur

r

pivot columns chodta hai; dono pivot count

r

ke barabar hain.

A x = b

solvable hai iff?

b \in C (A)

(the column space).

R^{n}

mein kaun se do subspaces orthogonal hain?

N (A) ⊥ C (A^{T})

, aur milke

R^{n}

banate hain.

N (A) ⊥ C (A^{T})

kyun hai?

A x = 0

matlab har row dot

x

zero hai, toh

x ⊥

har row, isliye

⊥

row space bhi.

Ek matrix ka trivial null space hota hai iff?

Uske columns linearly independent hain (

r = n

4 \times 6

matrix of rank 3: left null space ki dim?

m - r = 4 - 3 = 1