Foundations — Span — definition

4.5.16 · D1 · Maths › Linear Algebra (Full) › Span — definition

Parent note padhne se pehle, tumhe har woh symbol khud se samajh aana chahiye jo woh use karta hai. Neeche, har symbol zero se build kiya gaya hai — pehle plain words mein, phir ek picture, phir yeh bhi ki topic us symbol ke bina kyun nahi chal sakta. Koi bhi cheez define hone se pehle use nahi ki gayi hai.

0. Vector kya hota hai? (sabse pehla arrow)

Picture karo. Upar wala number horizontal hai ("kitna across"), neeche wala vertical ("kitna upar"). Arrow origin se us tip tak draw kiya jaata hai.

Topic ko iske kyun zaroorat hai. Span "har woh jagah hai jo arrows se reach ho sake." Toh hume pehle jaanna chahiye ki ek akela arrow kya hota hai aur use kaise name karte hain. Baaki sab kuch inhi se banta hai.

1. Origin aur zero vector $0$

Picture karo. Page ke centre mein ek dot. "Ghar par rehna" — tum kabhi nikle hi nahi.

Topic ko iske kyun zaroorat hai. Parent note kehta hai "span hamesha origin se guzarta hai." Woh statement $0$ ke baare mein hai: kyunki tumhe hamesha na hilne ki permission hai, zero vector hamesha reachable hai. Tum woh sentence tab tak nahi samajh sakte jab tak yeh symbol tumhara nahi ho jaata.

2. Scalars, letter $c$, aur $\mathbb{R}$

Picture karo. Ek number line jo left aur right mein hamesha ke liye failti hai. $c$ choose karna matlab us par kahin bhi pin lagana.

Topic ko iske kyun zaroorat hai. "Scale, flip, add" — scale aur flip bilkul "scalar se multiply karo" hai. Kyunki $c$ tamam $\mathbb{R}$ (negatives aur zero samet) par range karta hai, tumhara arrow kuch bhi nahi tak shrink ho sakta hai, peeche flip ho sakta hai, ya kisi bhi length tak stretch ho sakta hai. Parent ka mistake box "span ke liye sirf positive scalars chahiye" galat hai bilkul isliye kyunki $\mathbb{R}$ mein negatives shamil hain.

3. Scalar multiplication $cv$ — ek arrow ko stretch karna

Picture karo (tamam cases — yeh important part hai).

• $c>1$: arrow longer banta hai, same direction.
• $c=1$: koi change nahi.
• $0<c<1$: origin ki taraf shrink hota hai, same direction.
• $c=0$: $0$ (origin) par collapse hota hai.
• $c<0$: opposite direction mein flip hota hai, phir $|c|$ se scale hota hai.

Topic ko iske kyun zaroorat hai. Jaise $c$ tamam $\mathbb{R}$ par sweep karta hai, $cv$ ki tip origin se guzarne wali poori ek line trace karti hai. Yeh akela fact yahi kaaran hai ki "ek vector ka span ek line hoti hai." Figure mein yellow tip ko dekhte hao ki woh $c$ badalne par line par slide karti hai.

4. Vector addition $u+v$ — tip-to-tail

Picture karo. $u$ draw karo. Phir $v$ ko $u$ ki tip par shuru karo (ise rotate kiye bina slide karo). Jahan $v$ khatam hota hai woh $u+v$ ki tip hai. Yeh tip-to-tail rule hai.

Kyun yeh tool aur koi doosra nahi? Humein do arrows ko ek reachable spot mein combine karne ka tarika chahiye. Coordinate-wise addition woh akela rule hai jo physical "$u$ ke saath chalo, phir $v$ ke saath chalo" motion se match karta hai — parallelogram ka diagonal. Scaling length badalta hai; addition directions combine karta hai. Span ko dono chahiye, kyunki sirf ek move se tum stuck reh jaate.

5. Linear combination — scale, flip, add, sab ek saath

Picture karo. Har arrow ko uske apne scalar se stretch karo, phir unhe tip-to-tail chain karo. Final tip span mein ek point hai. Scalars badlo → kisi nayi jagah par utro.

Topic ko iske kyun zaroorat hai. Yahi span ka raw material hai. Parent ki definition $\mathrm{span}\{v_1,\dots ,v_k\}=\{c_1{v}_1+\cdots +c_k{v}_k\mid c_i\in \mathbb{R}\}$ literally "tamam linear combinations ka set" hai. Iska apna deep dive dekhne ke liye Linear combination refer karo.

6. Set-builder notation $\{\cdot \mid \cdot \}$

Picture karo. Ek cookie-cutter (recipe $cv$) jo $c$ badalne par infinitely many cookies stamp karta hai. Cookies ki poori pile woh set hai.

Topic ko iske kyun zaroorat hai. Span ek infinite collection hai. Braces yahi kehne ka tarika hai ki "tamam yeh ek saath" bina unhe list kiye. Parent ki mistake "span sirf $\{v_1,\dots ,v_k\}$ hai" braces ke andar thode se generators ko infinite recipe-output se confuse karti hai jo braces describe karte hain.

7. Subspace — woh shape jo span ban jaata hai

Picture karo. Origin se guzarne wali ek line, ya origin se guzarne wala ek plane — koi floating line nahi jo centre ko miss kare.

Topic ko iske kyun zaroorat hai. Yeh "span kaisa dikhta hai?" ka answer hai. Poori detail Subspace mein hai; yahan tumhe sirf teen-rule checklist chahiye taaki parent ki mini-derivation samjh aaye.

8. Determinant $\det$ — "kya yeh plane fill karte hain?" wala number

Topic ko iske kyun zaroorat hai. Parent ka aakhri worked example $\det =-5\ne 0$ use karta hai yeh conclude karne ke liye ki do vectors ${\mathbb{R}}^2$ span karte hain. Ab tum jaante ho ki nonzero number ka matlab "har jagah pahuncho" kyun hota hai.

Prerequisite map

Equipment checklist

Khud ko test karo — sirf zor se jawab dene ke baad reveal karo.

Connections

• Linear combination — yahan build ki gayi recipe har span ka building block ban jaati hai.
• Subspace — woh shape jo span hamesha ban jaata hai.
• Determinant — woh number jo "poora plane fill karta hai" detect karta hai.
• Linear independence — decide karta hai ki extra arrows nayi directions add karte hain ya nahi (next deep dive).
• Basis · Dimension · Column space — yeh foundations jahan le jaati hain jab span master ho jaata hai.