Subspaces — four fundamental subspaces of a matrix: Visual walkthrough

4.5.44 · D2 · Maths › Linear Algebra (Full) › Subspaces — four fundamental subspaces of a matrix

Step 0 — "Vector" kya hota hai aur " $A x$ " kya hai? (kuch assume nahi)

Hamara matrix $A$ ek input arrow $x = [x_{1} x_{2}]$ leta hai aur ek output arrow produce karta hai. Rule hai columns ko combine karo:

the machine A input arrow x = x_{1} column a_{1} [12] + x_{2} column a_{2} [24] .

$x_{1}$ = pehle column arrow ka kitna hissa lena hai.
$x_{2}$ = doosre column arrow ka kitna hissa lena hai.
Output do scaled arrows ko head-to-tail rakhne par milne wali tip hai.

$A x$ ko is tarah kyun padhein? Kyunki yeh ek abstract "matrix times vector" ko do fixed arrows ko mix karne ki picture mein badal deta hai. Wahi ek idea neeche har subspace generate karta hai.

Figure — Subspaces — four fundamental subspaces of a matrix

Step 1 — Do columns draw karo aur unhe dekho

WHAT we do: $a_{1} = [12]$ aur $a_{2} = [24]$ plot karo.

WHY: machine ka output hamesha inhi do ka mix hota hai, isliye reachable outputs ka set poori tarah decide hota hai ki yeh do arrows kahan point karte hain.

PICTURE: notice karo $a_{2} = 2 a_{1}$ — column 2, column 1 ko 2 se stretch kiya hua hai. Yeh dono ek hi line par hain. Toh inhe mix karne se kabhi us line se bahar nahi nikloge. Blue line hi poora column space hai: origin se guzarti ek 1-dimensional line, poora plane nahi.

Step 2 — Kaun se outputs FORBIDDEN hain? (the red zone)

WHAT: pucho "kya output kabhi blue line se bahar ja sakta hai?" Nahi. Koi bhi $b$ blue line se bahar kabhi $A x$ ki tarah nahi likha ja sakta.

WHY it matters: yeh exactly wahi statement hai ki "$Ax=b$ solvable hai $⟺ b \in C (A)$ ". Line se bahar ke target ka koi recipe nahi hota.

PICTURE: plane do coloured regions mein split hoti hai — reachable outputs ki blue line, aur baki sab kuch (red drawn) jo unreachable hai. Output space $R^{2}$ do mein cut ho jaata hai.

Step 3 — Kaun se inputs ZERO tak CRUSH ho jaate hain? (the null space)

WHAT we do: $A x = 0$ solve karo. Kyunki dono rows $[12]$ hain (row 2 $= 2 \times$ row 1), hume sirf chahiye:

coeff of x_{1} 1 x_{1} + coeff of x_{2} 2 x_{2} = 0 ⟹ x_{1} = - 2 x_{2} .

$x_{2} = 1$ pick karo: special solution hai $[- 2 1]$ .

WHY: do unknowns mein ek equation ek free variable chhodti hai, isliye $dim N (A) = n - r = 2 - 1 = 1$ — crushed inputs ki ek line. Dekho Rank–Nullity Theorem.

PICTURE (input space mein): $[- 2 1]$ se guzarti green line. Us par har arrow, machine mein daalo, zero arrow ke roop mein baahir aata hai.

Crush ka quick check: $A [- 2 1] = [1 (- 2) + 2 (1) 2 (- 2) + 4 (1)] = [00] . ✓$

Step 4 — Rows, aur yeh crush ke saath perpendicular kyun hain

WHAT we do: row arrow $r_{1} = [12]$ (orange) ko usi input plane mein draw karo jahan green null line hai.

WHY the tool "dot product": $A x = 0$ ka matlab hai har row dotted with $x$ zero ho. Dot product $r \cdot x = r_{1} x_{1} + r_{2} x_{2}$ bilkul ek sawaal ka jawaab deta hai — "kya yeh do arrows perpendicular hain?" — kyunki yeh zero hota hai exactly jab woh $9 0^{\circ}$ par milte hain. Isliye dotting, koi aur operation nahi, yahan geometry ko expose karta hai.

row dotted with input r_{1} \cdot x = = - 2 + 2 (1) (- 2) + (2) (1) = 0 \Rightarrow orange ⊥ green.

PICTURE: orange row line aur green null line ek perfect right angle par cross karte hain. Yeh $N (A) ⊥ C (A^{T})$ visible form mein hai. Dekho Orthogonal Complements.

Step 5 — Chautha subspace: forbidden outputs, transpose karke mila

WHAT we do: $A^{T} y = 0$ solve karo. $A$ ke columns, $A^{T}$ ki rows hain; dono $[12]$ ke multiples hain, jo ek equation deta hai:

from a_{1} 1 y_{1} + from a_{2} 2 y_{2} = 0 \Rightarrow y = [- 2 1] .

WHY: yeh direction har column ke perpendicular hai, isliye poori blue column line ke perpendicular hai. Yeh red forbidden zone ko name karta hai: outputs blue par pile up hote hain, aur woh direction jise aap kabhi reach nahi kar sakte exactly yahi red arrow hai.

PICTURE (output space mein): blue column line aur red left-null line $9 0^{\circ}$ par cross karte hain. $N (A^{T}) ⊥ C (A)$ , aur $dim N (A^{T}) = m - r = 2 - 1 = 1$ .

Orthogonality check (column $⊥$ left-null): $[12] \cdot [- 2 1] = - 2 + 2 = 0. ✓$

Step 6 — Edge case: full-rank machine (kuch crush nahi hota)

WHAT: $B = [2113]$ par switch karo, jiske columns alag directions mein point karte hain.

WHY show this: yeh prove karne ke liye ki upar ki pictures ek special (rank-deficient) case thin. Jab columns independent hote hain, story ka shape badal jaata hai aur aapko purani intuitions carry over nahi karni chahiye.

PICTURE: do blue columns ab poora plane span karte hain. Kuch bhi forbidden nahi hai ( $C (B) = R^{2}$ , red zone gayab). Zero tak crush hone wala akal input origin itself hai, isliye green null "line" ek single point tak shrink ho jaati hai.

r = 2 = m = n \Rightarrow dim N (B) = n - r = 0, dim N (B^{T}) = m - r = 0.

Ek-picture summary

Do planes side by side. Left = input space $R^{2}$ : orange row line aur green null line $9 0^{\circ}$ par, use fill karte hue. Right = output space $R^{2}$ : blue column line aur red left-null line $9 0^{\circ}$ par, use fill karte hue. Machine $A$ beech mein arrows karti hai, green $\to$ origin (crush) aur baki sab $\to$ blue line bhej ke.

Recall Feynman: poora walkthrough retell karo

Maine machine ke do columns draw kiye aur notice kiya ki woh ek line par hain — isliye jo outputs woh bana sakti hai (blue) sirf ek line form karte hain, aur us line se bahar sab kuch (red) forbidden juice hai. Phir maine pucha ki machine kaun se inputs flatten karti hai: $A x = 0$ solve karne se ek green line of "wasted recipes" mili. Maine machine ki rows (orange) usi input plane mein draw ki aur ek row ko ek null vector se dot kiya — woh zero nikla, matlab orange aur green ek right angle par cross karte hain aur mil ke input plane fill karte hain. Transposing ne aakhri piece diya: ek red direction blue outputs ke perpendicular, exactly forbidden zone ko name karta hua. Aakhir mein maine ek aisi matrix daali jiske columns spread apart hain — ab kuch bhi forbidden nahi aur kuch bhi crush nahi hota, yeh dikhate hue ki pehle wali squeeze special rank-1 case thi. Do lines in, do lines out, har pair perfect right angles par.

Recall

A x

x_{1} a_{1} + x_{2} a_{2}

ki tarah padhne se aap kya dekh sakte ho?

Output fixed column arrows ka ek mixture hai, isliye reachable outputs = span of columns =

C (A)

Hamare matrix

A = [1224]

mein,

C (A)

sirf ek line kyun hai?

Column 2

= 2 \times

column 1, isliye dono columns ek hi direction share karte hain; rank

= 1

Ek row ko

x

se dot karna

N (A)

ki geometry kyun expose karta hai?

(A x)_{i} = r_{i} \cdot x

, aur dot product zero hota hai exactly jab arrows perpendicular hoon — isliye

A x = 0

ka matlab hai

x ⊥

har row.

Jab

A

full rank ho (

r = n

) toh null space ka kya hota hai?

Woh single point

{0}

tak collapse ho jaata hai; kuch bhi crush nahi hota.

R^{m}

mein "forbidden output" direction ko kaun sa subspace name karta hai?

Left null space

N (A^{T})

, column space ke perpendicular.

Connections

Rank of a Matrix — woh ek number $r$ jisne in pictures mein har dimension fix kiya.
Rank–Nullity Theorem — green line ki dimension $n - r$ seedhi yahan se aayi.
Orthogonal Complements — Steps 4 aur 5 mein right angles.
Solving Ax=b — Step 2 ka red forbidden zone exactly unsolvable targets hai.
RREF and Pivots — drawings ke peeche pivot/free split kaise mili.
Least Squares & Projections — jab aapka target red zone mein land kare tab kya karein.