Span — definition — Hinglish notes

4.5.16 · Maths › Linear Algebra (Full)

Span KYA hai?

Yeh definition KYU? Kyunki ek vector space tumhe jo do operations deta hai woh hain scaling ( $c v$ ) aur adding ( $u + v$ ). Ek linear combination $c_{1} v_{1} + \dots + c_{k} v_{k}$ bas yahi do operations baar baar apply karne se banta hai. Span isliye closure hai — har woh cheez jo tum sirf allowed tools se bana sakte ho.

Span ko KAISE compute karte / sochte hain?

Yeh check karne ke liye ki "kya vector $w$ , $span {v_{1}, \dots, v_{k}}$ mein hai?", tum yeh poochte ho: $Kya aise scalars c_{i} exist karte hain ki c_{1} v_{1} + \dots + c_{k} v_{k} = w ?$ Yeh $c_{i}$ unknowns mein bas ek system of linear equations hai. Agar solution milta hai → haan; nahi toh → nahi.

Span kyun ek subspace hai (scratch se mini-derivation)

Hum claim karte hain ki $S = span {v_{1}, \dots, v_{k}}$ ek subspace hai. Humein dikhana hoga: (1) $0$ contain karta hai, (2) addition ke under closed hai, (3) scaling ke under closed hai.

Zero: saare $c_{i} = 0$ lo, jisse $0 \in S$ milta hai. ✔ Yeh step kyun? Saari-empty combination ek valid combination hai.
Addition: maano $x = \sum a_{i} v_{i}$ aur $y = \sum b_{i} v_{i}$ . Tab $x + y = \sum (a_{i} + b_{i}) v_{i},$ jo phir se ek linear combination hai → $S$ mein. ✔ Kyun? Humne like terms group kiye; $(a_{i} + b_{i})$ bas ek aur scalar hai.
Scaling: $λ x = λ \sum a_{i} v_{i} = \sum (λ a_{i}) v_{i} \in S$ . ✔ Kyun? $λ$ ko andar laane par bhi ek linear combination hi rehta hai.

Toh har span ek subspace hota hai — yahi woh gehra karan hai ki spans matter karte hain.

Worked examples

Common mistakes (Steel-manned)

Recall Feynman: ek 12-saal ke bacche ko explain karo

Tumhare paas kuch magic arrows hain. Tum har arrow ko lamba ya chota kar sakte ho, use ulta point kar sakte ho, aur arrows ko tip-to-tail jod sakte ho. Span woh har jagah hai jahan tum in arrows se ja sakte ho. Agar tumhare arrows sach mein alag directions mein point karte hain, toh tum ek poori sheet (plane) fill kar sakte ho. Agar woh secretly ek hi direction mein point karte hain, toh tum sirf ek line par travel kar sakte ho. Aur tum hamesha ghar reh sakte ho (origin par) bilkul na hilkar.

Active recall

Vectors ke ek set ka span kya hota hai?

Saare linear combinations

c_{1} v_{1} + \dots + c_{k} v_{k}

ka set jahan scalars

c_{i} \in R

hain.

Har span mein zero vector kyun hota hai?

Saare scalars

c_{i} = 0

choose karne par

0

milta hai, jo hamesha ek valid linear combination hai.

Kya span hamesha ek subspace hota hai?

Haan — ismein

0

hai aur yeh addition aur scalar multiplication ke under closed hai.

Span banane mein kaun si do operations kaam aati hain?

Scalar multiplication (scale/flip) aur vector addition.

w \in span {v_{i}}

hai ya nahi, yeh kaise test karte hain?

Linear system

\sum c_{i} v_{i} = w

solve karo; agar solution exist karta hai, toh

w

span mein hai.

Kya do vectors hone se guarantee hoti hai ki woh plane span karte hain?

Nahi — sirf tab jab woh linearly independent hon; agar ek doosre ka multiple ho toh sirf ek line span hoti hai.

Linear combination mein kaun se scalars allowed hain?

Saare real numbers, negatives aur zero bhi shamil hain (sirf positive nahi).

Naya vector add karne par span kab badata hai?

Sirf tab jab naya vector already current span mein nahi hota (ek naya independent direction).

Geometrically,

R^{3}

mein span kaisa dikh sakta hai?

Ek point (sirf origin), ek line, ek plane, ya poora

R^{3}

— hamesha origin se guzarta hua.

det \neq = 0

tumhe

v_{1}, v_{2}

R^{2}

span karne ke baare mein kya batata hai?

Nonzero determinant ⇒ matrix invertible hai ⇒ har vector reachable hai ⇒ woh

R^{2}

span karte hain.

Connections

Linear combination — har span ka building block.
Linear independence — control karta hai ki extra vectors span badhate hain ya nahi.
Subspace — har span ek hota hai; spans subspaces produce karne ka standard tarika hain.
Basis — ek minimal spanning set (spans + independent).
Dimension — basis mein vectors ki sankhya = span ka "size".
Column space — ek matrix ke columns ka span.
Determinant — square cases mein full-span test karta hai.

Span — definition