1.1.10
Ek matrix A machine hai jo vectors ko transform karti hai : x ↦ A x
Kaunse outputs produce kar sakti hai? → column space (uski pahunch).Kaun se inputs zero ho jaate hain? → null space (uska andha kona).
Rank un sachchi independent directions ki ginti hai jo machine zinda rakhti hai. Solvability, invertibility, aur information loss ke baare mein baaki sab kuch inhi teen ideas se nikalta hai.
A ∈ R m × n a 1 , … , a n Col ( A ) = { A x : x ∈ R n } = span { a 1 , … , a n } ⊆ R m . sabhi vectors b A x = b .
YE SPAN KYU BARABAR HAI: A x = x 1 a 1 + x 2 a 2 + ⋯ + x n a n columns ka linear combination hai. x
Definition Null space (kernel)Null ( A ) = { x ∈ R n : A x = 0 } ⊆ R n . zero par map karti hai — woh directions jinhe woh bhool jaati hai.
rank ( A ) = dim ( Col ( A )) = number of independent columns = number of pivots .
Intuition Row reduction kyu kaam karta haiRow operations invertible left-multiplications E E A linear dependence relations nahi badlate, aur yeh Null ( A ) nahi badlate (kyunki E A x = 0 ⟺ A x = 0
Procedure:
A R Pivot columns = leading 1's ki positions.rank = Col ( A ) = original A pivot columns ka span (na ki R Null ( A ) special solutions nikalo; yeh ek basis banate hain.
Common mistake Steel-man: "Col(A) = R ke pivot columns ka span"Yeh sahi kyun lagta hai: R e 1 , e 2 lagte hain.
Yeh galat kyun hai: Row operations columns ko entry-wise badal dete hain, isliye Col ( R ) = Col ( A ) R kaun se columns independent hain; phir aap unhe original A se lete ho.
Fix: R positions dhundne ke liye karo; corresponding columns A
Scratch se Derivation. A n pivot column hai ya free column .
Pivot columns ki sankhya = rank ( A )
Har free column ek free variable deta hai, aur har free variable bilkul ek special solution produce karta hai — Null ( A ) 1
Isliye dim ( Null ( A )) =
Kyunki har column ya pivot ya free hai, bilkul ek baar:
rank ( # pivots ) + d i m Null ( # free ) = total columns n . ■
Worked example Example 1 — poora pipelineA = 1 2 3 2 4 6 1 3 4 Row reduce karo. R 2 → R 2 − 2 R 1 R 3 → R 3 − 3 R 1 1 0 0 2 0 0 1 1 1 R 3 − R 2 1 0 0 2 0 0 1 1 0 R 1 − R 3 1 0 0 2 0 0 0 1 0 = R . Yeh step kyun? Hum RREF pahunchne ke liye pivots ke neeche aur upar clear karte hain, pivot columns 1 aur 3 expose karte hue.
rank = 2 Col(A) = original A ke columns 1 aur 3 ka span: span {( 1 , 2 , 3 ) T , ( 1 , 3 , 4 ) T } Kyun? Column 2 = 2 × Null(A): free variable x 2 = t R x 1 + 2 x 2 = 0 ⇒ x 1 = − 2 t x 3 = 0 x = t ( − 2 , 1 , 0 ) T Null ( A ) = span {( − 2 , 1 , 0 ) T } Rank–nullity check: 2 + 1 = 3 = n
Worked example Example 2 — ek "wide" matrix, guaranteed null spaceA = [ 1 0 0 1 2 3 ] ( m = 2 , n = 3 ) . = 2 x 3 = t x 1 = − 2 t , x 2 = − 3 t Null = span {( − 2 , − 3 , 1 ) T } = 1 Yeh kyun important hai: n > m ≤ m < n dim Null ≥ n − m > 0 hamesha nonzero null space hota hai (infinitely many inputs collapse ho jaate hain). Yeh step kyun? Rank rows ki sankhya se zyada nahi ho sakti, isliye free columns force hote hain.
Worked example Example 3 — ek ML design matrix ka Feynman-style readMaan lo data matrix X ∈ R 100 × 5 4 multicollinearity ). dim Null ( X ) = 5 − 4 = 1 w = 0 X w = 0 w w + c w identical predictions dete hain → least-squares solution unique nahi hai . Yeh step kyun? X X w
Recall Compute karne se pehle predict karoB = [ 1 1 1 1 ]
Forecast: columns identical hain → rank 1, nullity 2 − 1 = 1
Verify: RREF = [ 1 0 1 0 ] span {( − 1 , 1 ) T }
A A b A x = b R m
A Woh sab x A x = 0 R n
Rank ko teen equivalent tareekon se define karo. (1) dim Col ( A )
Rank–nullity theorem batao. rank ( A ) + dim Null ( A ) = n
Rank–nullity kyu hold karta hai? Har column ya toh pivot (→rank) ya free (→ek null-space basis vector) hai; yeh sab n
Col(A) ke liye basis dhundhne ke liye kaun se columns lete ho? ORIGINAL A
Col(R), Col(A) ke barabar kyun nahi hota? Row operations columns ko entry-wise badal dete hain; yeh dependence relations preserve karte hain lekin column space khud nahi.
Ek m × n n > m rank ≤ m < n ≥ n − m > 0
Kya row operations null space badlate hain? Nahi: E A x = 0 ⟺ A x = 0 E
A x = b Tab hi jab b ∈ Col ( A )
Ek square n × n Full rank n Null ( A ) = { 0 }
ML: design matrix X Multicollinearity → non-unique least-squares weights (w X w = 0
Recall Feynman: 12 saal ke bachche ko samjhaoEk photocopier imagine karo jo sirf do ink colors mila ke copies bana sakti hai. Column space woh har woh color hai jo woh print kar sakti hai — uski poori palette. Rank kitni sachchi alag alag inks hain (agar red aur pink basically same hain, toh yeh actually ek ink hai, do nahi). Null space woh ink amounts ki secret recipe hai jo bilkul blank page print kari — tune ek ki positive aur doosre ki negative mili toh woh cancel ho gayi. Agar machine ke paas knobs se kam real inks hain, toh hamesha blank-page recipes hongi: kaafi alag alag knob settings same picture deti hain.
"Columns Reach, Null Forgets, Rank Keeps."
Aur theorem ke liye: P + F = C → P ivots + F ree = C olumns (rank + nullity = n
