Linear transformations — definition, kernel, image

4.5.27 · Maths › Linear Algebra (Full)

1. Definition

YE do rules kyun? Ek vector space define hota hai do operations se: vectors ko add karna aur unhe scale karna. Ek map "structure preserve" tab karta hai jab wo precisely un dono operations ke saath commute kare. Na zyada, na kam.

Definition ko kaise use karein (Derivation: $T (0) = 0$ ): $T (0) = T (0 \cdot v) = 0 \cdot T (v) = 0.$ Ye step kyun? Maine $0$ ko scalar $0$ times kisi bhi vector ke roop mein likha, phir scalar bahar nikaala (homogeneity). Isliye har linear map zero vector ko zero vector par bhejta hai — ek free sanity check.

2. Kernel (null space)

Ye subspace kyun hai? Agar $T (u) = 0$ aur $T (v) = 0$ toh $T (c u + d v) = c T (u) + d T (v) = 0$ . Toh kernel vectors ke sums aur scalings kernel mein hi rehte hain → ye $V$ ka ek subspace hai (aur hamesha $0$ contain karta hai).

3. Image (range)

$W$ ka subspace kyun hai? Har output $T (v)$ hai; $c T (u) + d T (v) = T (c u + d v)$ phir bhi ek output hai. Operations ke under closed hai → subspace.

Figure — Linear transformations — definition, kernel, image

4. Rank–Nullity Theorem (bridge)

Scratch se Derivation. Maano $dim V = n$ . $ker T$ ka ek basis lo ${k_{1}, \dots, k_{p}}$ (toh nullity $= p$ ). Ise $V$ ke full basis tak extend karo: ${k_{1}, \dots, k_{p}, w_{1}, \dots, w_{q}}$ jahan $p + q = n$ ho. Claim: ${T (w_{1}), \dots, T (w_{q})}$ , $im T$ ka basis hai.

Ye image span karte hain: koi bhi $v = \sum a_{i} k_{i} + \sum b_{j} w_{j}$ , toh $T (v) = \sum a_{i} T (k_{i}) + \sum b_{j} T (w_{j}) = \sum b_{j} T (w_{j})$ (kernel terms vanish ho jaate hain). Kyun? $T (k_{i}) = 0$ kernel ki definition se.
Ye independent hain: agar $\sum b_{j} T (w_{j}) = 0$ toh $T (\sum b_{j} w_{j}) = 0$ , toh $\sum b_{j} w_{j} \in ker T = span (k_{i})$ . Ise $\sum c_{i} k_{i}$ ke roop mein likhne aur basis independence use karne se sabhi $b_{j} = 0$ force hote hain. Toh rank $= q$ , isliye $n = p + q =$ nullity $+$ rank. $■$

Apne example par check: $dim V = 3$ , nullity $= 1$ , rank $= 2$ , aur $1 + 2 = 3$ . ✓

5. Common galtiyan

6. Active recall

Recall Quick self-test (answers chhupaao)

Ek linear map ki do defining properties batao. → additivity & homogeneity.
$T (0) = 0$ kyun hona chahiye? → $T (0 v) = 0 T (v) = 0$ .
$ker T \subseteq$ ? → $V$ . $im T \subseteq$ ? → $W$ .
Injective iff? → $ker T = {0}$ .
Rank–nullity statement? → $dim V =$ nullity $+$ rank.

Recall Feynman: ek 12-saal ke bachche ko samjhao

Ek machine imagine karo jo arrows leta hai aur naye arrows deta hai, ek fair rule ke saath: agar tum pehle do arrows add karo phir machine mein daalo, toh same answer milta hai jaise dono ko alag daalne ke baad add karo (aur ek arrow ko double karne se output bhi double hota hai). Kernel un arrows ka dhera hai jise machine bilkul납 flat karke zero kar deti hai. Image un sabhi arrows ka collection hai jo machine produce kar sakti hai. Agar sirf "kuch nahi karo" waala arrow zero hota hai, toh machine kabhi do alag arrows ko confuse nahi karta. Counting rule: (jo crush ho gaye) + (jo ban sakte hain) = (jo shuru mein the).

Flashcards

Linear transformation ko define karne waali do properties kya hain?

Additivity

T (u + v) = T (u) + T (v)

aur homogeneity

T (c v) = c T (v)

Har linear map

T (0) = 0

kyun satisfy karta hai?

T (0) = T (0 \cdot v) = 0 \cdot T (v) = 0

homogeneity se.

T : V \to W

ke kernel ko define karo.

ker T = {v \in V : T (v) = 0_{W}}

V

ka ek subspace.

T : V \to W

ke image ko define karo.

im T = {T (v) : v \in V}

W

ka ek subspace.

T

injective hai if and only if ___?

ker T = {0}

T

surjective hai if and only if ___?

im T = W

Rank–nullity theorem batao.

dim V = dim (ker T) + dim (im T)

= nullity + rank.

T (x) = A x

ke liye, image kaunsi familiar space ke barabar hai?

A

ka column space.

Kernel subspace kyun hai?

Agar

T (u) = T (v) = 0

toh

T (c u + d v) = c T (u) + d T (v) = 0

, combinations ke under closed hai.

T (x) = A x + b

(

b \neq = 0

ke saath) linear kyun nahi hai?

T (0) = b \neq = 0

T (0) = 0

violate hota hai.

Connections

Matrix multiplication — har $T : R^{n} \to R^{m}$ apne matrix $A = [T (e_{1}) \dots T (e_{n})]$ se multiplication hai.
Column space and null space — image = column space, kernel = null space.
Rank–Nullity Theorem — dimension accounting ka bridge.
Injective surjective bijective — kernel/image characterizations.
Change of basis — $T$ ka matrix kaise transform hota hai.
Eigenvalues and eigenvectors — $ker (T - λ I)$ eigenspace hai.