Rank of a matrix — definition, row rank = column rank theorem

4.5.12 · Maths › Linear Algebra (Full)

Rank KYA hai?

Teen tarike se define kyun kiya? Har roop alag situation mein useful hai:

Row space batata hai equations / constraints ke baare mein.
Column space batata hai ki $A x$ ke kaunse outputs reachable hain.
Pivots rank nikalne ka ek computational tarika dete hain.

Rank KAISE compute karein (pehle worked example, theory baad mein)

Row rank = column rank KYU hota hai? (Scratch se derivation)

Maano $r$ = $A$ ( $m \times n$ ) ka column rank hai. Hum prove karenge row rank $\leq r$ , phir symmetry se equality milegi.

Step 1 — column basis chuno. Maano ${c_{1}, \dots, c_{r}}$ column space ka basis hai. $C = [c_{1} \dots c_{r}]$ banao, jo ek $m \times r$ matrix hai.

Kyun? $A$ ka har column in $r$ vectors ka combination hai — yahi "column space ka basis" ka matlab hai.

Step 2 — $A = C R$ factor karo. Har column $A_{j}$ , $C$ times ek coefficient vector ke barabar hai. Un coefficient vectors ko $r \times n$ matrix $R$ ke columns ki tarah stack karo. Tab $A = C R, C : m \times r, R : r \times n .$

Yeh key trick kyun hai? Humne $A$ ko ek product ke roop mein likha jahan beech ki dimension exactly column rank $r$ hai. Yeh factorization rows ke baare mein bhi jaankari "leak" karne par majboor hai.

Step 3 — rows padho. $A$ ki row $i$ , $C$ ki row $i$ times $R$ hai: $(row i of A) = (row i of C) R .$ Toh $A$ ka har row, $R$ ke $r$ rows ka linear combination hai.

Yeh kaise khatam karta hai? $A$ ke rows sirf $r$ vectors ke span mein rehte hain ( $R$ ke rows). Isliye $row rank (A) \leq r = column rank (A) .$

Step 4 — symmetry. Wahi argument $A^{T}$ par apply karo. $A^{T}$ ke rows, $A$ ke columns hain aur vice versa, toh: $column rank (A) = row rank (A^{T}) \leq column rank (A^{T}) = row rank (A) .$ Dono inequalities combine karke: $row rank (A) = column rank (A) = rank (A)$

Figure — Rank of a matrix — definition, row rank = column rank theorem

Pehle Forecast karo, phir Verify karo

Common mistakes (Steel-manned)

Key bounds (derive karo, memorize mat karo)

Recall Feynman: 12 saal ke bacche ko samjhao

Socho matrix LEGO instructions ki ek list hai, har row batata hai ek cheez kaise banate hain. Kuch instructions sirf copies ya mixes hain doosron ki — ve kuch naya nahi sikhate. Rank yeh hai ki kitni genuinely nayi instructions hain. Jaadoo? Chahe tum "nayi instructions" rows dekh ke gino YA columns dekh ke gino, hamesha same number milta hai. Jaise ek room mein log ginne ke liye heads gino ya joote ke pair gino — tumhe trust hai ki match karenge, aur yahan hamesha karte hain.

Flashcards

Rank ki teen equivalent definitions kya hain?

dim(row space) = dim(column space) = echelon form mein pivots ki sankhya.

Rank ka central theorem batao.

Row rank = column rank har matrix ke liye; yeh common value hi rank hai.

Proof mein kaunsi factorization use hoti hai?

A = C R

jahan

C

m \times r

hai (column basis),

R

r \times n

hai; inner dimension

r

= rank.

A = C R

se row rank

\leq

column rank kyun prove hota hai?

A

ki row

i

= (

C

ki row

i

)·

R

, toh saare rows,

R

r

rows ka combination hain.

Row operations ke under rank invariant kyun hai?

Row ops rows ko rows ke linear combinations se replace karte hain, row space (aur uski dimension) kabhi nahi badlte.

m \times n

matrix ke rank par bound?

0 \leq rank (A) \leq min (m, n)

rank (A B)

bound?

\leq min (rank A, rank B)

Column space ka basis kahan se milta hai?

ORIGINAL matrix ke pivot positions wale columns se (reduced wale se nahi).

Rank aur addition ke baare mein common galat claim kya hai?

rank (A + B) = rank A + rank B

GALAT hai; sirf

\leq

hold karta hai.

Rank–nullity statement?

rank (A) + dim ker (A) = n

(columns ki sankhya).

Connections

Row Space and Column Space
Gaussian Elimination & Echelon Form
Rank–Nullity Theorem
Linear Independence and Basis
Matrix Multiplication as Linear Combination
Invertible Matrix Theorem (full rank ⇔ invertible for square $A$ )