4.5.13
Intuition Matrix ke baare mein do fundamental questionsEk matrix A x A x
Column space poochta hai: kaunse outputs produce kar sakta hoon main? (pahunchne wala set)Null space poochta hai: kaunse inputs zero ho jaate hain? (wo "kho jaane wali" directions)
Har linear map ke ye do saaye hote hain. Inhe samajhna = map ko samajhna.
Definition Column space (image)A ∈ R m × n column space hai
Col ( A ) = { A x : x ∈ R n } ⊆ R m . A output space R m
Definition Null space (kernel)Null space hai
Null ( A ) = { x ∈ R n : A x = 0 } ⊆ R n . zero vector mein crush kar deta hai . Ye input space R n
Intuition Har ek subspace KYU hai?Ek set subspace hota hai agar usme 0
Null ( A ) A x = 0 A y = 0 A ( α x + β y ) = α ⋅ 0 + β ⋅ 0 = 0 Col ( A ) A x , A y α A x + β A y = A ( α x + β y )
Linearity dono ko majboor karti hai ki wo origin se guzarne wali flat sheets banen — kabhi curved nahi, kabhi offset nahi.
Sab kuch row reduction to RREF se aata hai. A R nahi badalta (row operations A x = 0 zaroor badalta hai — isliye hum columns original A
Definition Pivot vs free columnsRREF R pivot column mein ek leading 1 hota hai.
r = = rank ( A ) Pivot variables ↔ pivot columns. Free variables ↔ free columns.
Original columns KYU? Row reduction rows ko mix karta hai, column entries distort ho jaati hain, isliye generally Col ( R ) = Col ( A ) sahi batata hai ki kaunse columns independent hain — wo dependence relations exactly preserved rehte hain (wo null space mein rehte hain, jo unchanged hai). Isliye hum indices rakhte hain, lekin columns A
R x = 0 1 0 special solution deta hai.
Intuition YE sach KYU hona chahiyen pivot hota hai (genuinely nayi output direction contribute karta hai → rank mein count hota hai) ya free hota hai (ye dependent hai → zero mein collapse hone ki ek degree of freedom contribute karta hai → nullity mein count hota hai). Har column exactly ek baar count hota hai. Isliye dono numbers ka sum n
Worked example Example 1 — ek 3×4 matrixA = 1 3 1 2 6 2 2 6 3 4 12 6 Step 1: Row reduce.
R 2 → R 2 − 3 R 1 R 3 → R 3 − R 1
\begin{bmatrix}1&2&0&0\\0&0&1&2\\0&0&0&0\end{bmatrix}=R.$$
*Ye step KYU?* RREF pivots ko cleanly expose karta hai; pivots ke upar clear karne se free-variable formulas directly mil jaate hain.
**Step 2: Pivots** columns 1 aur 3 mein hain → $r=2$.
**Col(A) basis** = *original $A$* ke columns 1 aur 3: $\{(1,3,1)^\top,\ (2,6,3)^\top\}$. To $\dim\mathrm{Col}=2$.
*Original columns KYU?* $R$ ke mixed columns ab sahi output directions mein nahi point karte.
**Step 3: Null space.** Free variables $x_2,x_4$. $R$ se:
$x_1=-2x_2$, $x_3=-2x_4$.
- $x_2=1,x_4=0$: $s_1=(-2,1,0,0)^\top$.
- $x_2=0,x_4=1$: $s_2=(0,0,-2,1)^\top$.
**Null(A) basis** $=\{s_1,s_2\}$, $\dim\mathrm{Null}=2$.
**Check:** $r+\text{nullity}=2+2=4=n$. ✓ Aur indeed $A s_1 = -2(\text{col}_1)+(\text{col}_2)=0$. ✓
Worked example Example 2 — full column rank (trivial kernel)A = 1 0 1 0 1 1 . = 1 0 0 0 1 0 r = 2 = n Null ( A ) = { 0 } dim = 0 Col ( A ) = span {( 1 , 0 , 1 ) , ( 0 , 1 , 1 )} R 3 Full column rank ka matlab trivial kernel KYU hota hai? Koi free variable nahi ⇒ A x = 0 x = 0 injective hai.
Worked example Example 3 — rank jaldi padhna (80/20)A = [ 2 1 4 2 ] . = 2 × r = 1 dim Null = 2 − 1 = 1 2 x 1 + 4 x 2 = 0 ⇒ x 1 = − 2 x 2 ( − 2 , 1 ) ⊤ Ye fast KYU? Ek dependence spot karna full reduction ko skip karta hai — zyaadatar exam matrices mein ek obvious dependence hoti hai.
Common mistake "Col(A) ke liye RREF ke columns use karo."Ye sahi kyun lagta hai: RREF hi simplified matrix hai, to zaroor uske columns simplified column space hain. Ye galat kyun hai: row operations column entries ko scramble karte hain, output vectors ko move karte hain. RREF preserve karta hai ki kaunse columns independent hain, unki actual values nahi.
Fix: RREF se pivot positions identify karo, phir wo columns original A
Common mistake "Null spaceR m Ye sahi kyun lagta hai: tumne ise A m Ye galat kyun hai: null space vectors inputs x n R m Fix: Null ⊆ input space R n R m
Common mistake "Nullity = RREF mein zero rows ki sankhya."Ye sahi kyun lagta hai: zero rows 'khoyi hui' information jaisi lagti hain. Ye galat kyun hai: zero rows dependent rows count karti hain (row rank). Nullity free columns count karta hai = n − r Fix: nullity = n − ( number of pivots )
Recall Quick self-test (answers cover karo)
Col(A) ki dimension? → rank r
Null(A) ki dimension? → n − r
Kaunsa space R m
Col(A) basis ke columns kahan se lete hain? → original A
Kitne special solutions hote hain? → ek per free variable.
Recall Feynman: ek 12-saal ke bacche ko samjhaoEk vending machine imagine karo. Tum buttons dabate ho (ye tumhara input x
Column space wo hai — machine kabhi bhi jo snacks de sakti hai woh saare — shayad sirf chips aur soda stocked hain, to chahe kuch bhi press karo, chocolate nahi milegi.Null space wo hai — button combos jo tumhe kuch nahi dete — ye press karo aur machine bas tumhari mehnat kha jaati hai aur khaali haath wapas deti hai.
Counting rule: jo bhi button tum press karo wo ya to useful hai (naya snack deta hai) ya wasted hai (kuch naya nahi karta). Total buttons = useful + wasted. Ye hai rank + nullity = n
Mnemonic Ghar aur sizes yaad rakho"CO lumns O utputs hain (R m N ull iN puts hai (R n "Rank Plus Null = N" → r + ( n − r ) = n
Rank of a matrix — rank hi dim Col ( A ) Rank–Nullity Theorem — yahan use ki gayi counting identity.Row reduction & RREF — dono bases ka engine.Linear independence and basis — pivots ⇔ independent columns.Injective and surjective linear maps — trivial kernel ⇔ injective; full row rank ⇔ surjective.Four fundamental subspaces — Col(A), Null(A), Row(A), Null(Aᵀ).Solving Ax=b — solvable tabhi jab b ∈ Col ( A )
Definition of Col(A)? A { A x } R m
Definition of Null(A)? { x : A x = 0 } R n
Basis of Col(A)? ORIGINAL matrix A
Basis of Null(A)? Special solutions — ek per free variable, R x = 0
Dimension of Col(A)? Rank r
Dimension of Null(A) (nullity)? n − r
State Rank–Nullity. dim Col ( A ) + dim Null ( A ) = n r + ( n − r ) = n
Why derive Rank–Nullity? n
Why use original columns, not RREF columns, for Col(A)? Row ops column values badal dete hain; RREF sirf batata hai ki kaunse columns independent hain, unki actual output directions nahi.
When is Null(A)={0}? Jab koi free variable na ho, yaani full column rank r = n
Why is Null(A) a subspace? Agar A x = 0 , A y = 0 A ( α x + β y ) = 0
Row reduction effect on the two spaces? Null(A) preserve karta hai (A x = 0
pivot columns of original A
free variables give special solutions
