Subspaces — four fundamental subspaces of a matrix: Question bank

4.5.44 · D5 · Maths › Linear Algebra (Full) › Subspaces — four fundamental subspaces of a matrix

Shuru karne se pehle, do words jo hum baar baar use karte hain:

$A$ hai $m \times n$ — matlab $m$ rows (har output coordinate ke liye ek) aur $n$ columns (har input coordinate ke liye ek). To inputs $x$ mein $n$ entries hoti hain, outputs $A x$ mein $m$ entries hoti hain.
$r = rank (A)$ = row reduction ke baad pivots ki sankhya — dekho Rank of a Matrix aur RREF and Pivots.

Galti dhundho

Har line mein ek plausible-sounding claim hai jisme ek hidden mistake hai. Use naam do.

{(x, 2 x + 1) : x \in R}

ek subspace hai

R^{2}

ka kyunki yeh ek straight line hai."

Galat — yeh origin miss karta hai:

x = 0

par

(0, 1) \neq = 0

milta hai. Ek line ko zero-vector rule satisfy karne ke liye

0

se guzarna chahiye.

"Kyunki row rank = column rank hai, isliye row space column space ke barabar hai."

Equal dimension aur equal set ka confusion hai. Row rank

=

column rank

= r

counts ke baare mein statement hai; spaces alag dimensions mein hote hain jab tak

A

square na ho (aur tab bhi necessarily coincide nahi karte).

A

hai

4 \times 6

rank

3

ke saath, isliye

dim N (A^{T}) = 6 - 3 = 3

Galat subtraction — left null space

m

use karta hai:

dim N (A^{T}) = m - r = 4 - 3 = 1

. Plain null space

n

use karta hai:

dim N (A) = 6 - 3 = 3

C (A)

nikalne ke liye main RREF ke pivot columns leta hoon, kyunki woh independent hain."

Pivot columns original

A

ke lo, RREF ke nahi. Row operations column space badal deti hain, isliye RREF columns galat set span karte hain; sirf pivot positions transfer hoti hain.

N (A)

aur

C (A)

orthogonal complements hain."

Yeh alag spaces mein rehte hain (

R^{n}

R^{m}

) isliye complements nahi ho sakte. Sahi pairs hain

N (A) ⊥ C (A^{T})

(

R^{n}

mein) aur

N (A^{T}) ⊥ C (A)

(

R^{m}

mein).

A

ka rank

A

mein nonzero rows ki sankhya ke barabar hai."

Nahi — nonzero rows row reduction to echelon form ke baad count karo, raw matrix mein nahi. Original rows dependent ho sakti hain (jaise row 2

= 2 \times

row 1) aur phir bhi dono nonzero ho sakti hain.

"Agar

A

ke do columns identical hain, to rank drop hota hai lekin null space trivial rehta hai."

Contradiction — identical columns dependent hain, isliye ek nonzero combination

A x = 0

deta hai (un do columns par

+ 1, - 1

rakho). Woh vector

N (A)

mein hai, isliye

N (A) \neq = {0}

Edge cases

Boundary aur degenerate inputs — reader ko koi unseen scenario mat dikhne do.

Zero matrix

A = 0_{m \times n}

: uske chaaron subspaces kya hain?

r = 0

, isliye

C (A) = {0} \subseteq R^{m}

C (A^{T}) = {0} \subseteq R^{n}

N (A) = R^{n}

(sab kuch crush ho jaata hai),

N (A^{T}) = R^{m}

(kuch bhi reachable nahi hai).

Ek invertible

n \times n

matrix: chaaron describe karo.

r = n

, isliye

C (A) = C (A^{T}) = R^{n}

aur dono null spaces

{0}

hain. Kuch crush nahi hota, har output reachable hai — machine ek perfect bijection hai.

Ek single row vector

A = [123]

(so

1 \times 3

): har subspace kaisa dikhta hai?

r = 1

;

C (A) = R^{1}

(output space mein ek full line),

C (A^{T}) = span {(1, 2, 3)}

N (A)

plane

x_{1} + 2 x_{2} + 3 x_{3} = 0

hai (dim

2

N (A^{T}) = {0}

kyunki

m - r = 0

Ek single column vector

A = 123

(so

3 \times 1

): chaaron subspaces?

r = 1

;

C (A) = span {(1, 2, 3)}

(

R^{3}

mein ek line),

N (A) = {0}

(

n - r = 0

C (A^{T}) = R^{1}

N (A^{T})

R^{3}

mein woh plane hai jo

(1, 2, 3)

ke orthogonal hai, dim

m - r = 2

Kya null space aur left null space dono

{0}

ho sakte hain ek non-square matrix ke liye?

Nahi —

dim N (A) = n - r

aur

dim N (A^{T}) = m - r

dono

0

nahi ho sakte jab tak

r = m = n

na ho, jo ek square (invertible) matrix force karta hai. Ek rectangular matrix mein hamesha kam se kam ek nontrivial null space hota hai.

Ek wide matrix

m < n

jiska full row rank

r = m

hai: kaun sa subspace forced nontrivial hai?

N (A)

ki dimension

n - m > 0

hai, isliye infinitely many inputs kisi bhi reachable

b

par map karte hain; lekin

N (A^{T}) = {0}

aur

C (A) = R^{m}

, isliye har output reachable hai.

Ek tall matrix

m > n

jiska full column rank

r = n

hai: kaun sa subspace forced nontrivial hai?

N (A^{T})

ki dimension

m - n > 0

hai, isliye kuch outputs unreachable hain (least-squares territory); lekin

N (A) = {0}

, isliye koi bhi solvable

A x = b

unique hai.

Agar

r = 0

hai, to kya orthogonality pairings phir bhi valid hain?

Haan —

{0} ⊥ R^{n}

trivially aur yeh

R^{n}

tak sum karte hain; Fundamental Theorem is degenerate case mein bhi hold karta hai jahan ek partner poora space hota hai.

Recall Ek-line self-test

Answers cover karo aur "Spot the error" block mein race karo: agar tum har ek mein mistake das second se kam mein naam de sako, to tumne input/output ( $n$ vs $m$ ) aur dimension ( $r$ , $n - r$ , $m - r$ ) split internalize kar li hai.

Connections

Subspaces — four fundamental subspaces of a matrix — woh parent jise yeh bank drill karta hai.
Rank of a Matrix — woh number $r$ jis par har trap depend karta hai.
Rank–Nullity Theorem — $r + (n - r) = n$ ka bookkeeping.
Orthogonal Complements — kyun pairs perpendicular hain.
Solving Ax=b — solvability aur uniqueness traps.
RREF and Pivots — jahan se pivots (aur pivot-column error) aati hain.
Least Squares & Projections — left-null-space ka application.

Question bank — Subspaces — four fundamental subspaces of a matrix

True ya false — justify karo

Galti dhundho

Why questions

Edge cases

Connections