Basis — definition, uniqueness of representation: Foundations

4.5.17 · D1 · Maths › Linear Algebra (Full) › Basis — definition, uniqueness of representation

Is sentence par trust karne se pehle, tumhe us parent note ke har symbol ko apna banana hoga. Ye page har ek ko kuch nahi se build karta hai, ek aisi order mein jahan har piece apne pehle wale piece par tikti hai. Agar ek samajhdar 12-saal-ka baccha line one se padhe, toh use koi aisa symbol nahi milna chahiye jo usne pehle na dekha ho.

1. Vector actually hai kya? (symbol $v$ )

Ek vector koi bhi cheez hai jiske saath tum do kaam kar sako: do ko add karo, aur ek ko bada ya chhota scale karo. Bas itna hi kaam hai uska.

Jo picture ise concrete banati hai: ek arrow jo origin (point $(0, 0)$ ) se shuru hoke kisi jagah point karta hai.

Figure — Basis — definition, uniqueness of representation

$v_{1}$ jaisi entry ko component kehte hain — ek number jo batata hai ki ek axis ke along kitna door jana hai. Dhyan raho: ek component is baat se tied hai ki tumne axes kahan kheeche. Axes badlo, components badal jaate hain, chahe arrow wahi raha ho. Is baat ko yaad rakho — yeh Section 6 mein wapas aata hai.

2. Scale karne wala number: scalar (symbol $c$ )

Jab hum $c v$ likhte hain, toh letter $c$ ek simple number hai — ise scalar kehte hain kyunki iska kaam vector ko scale karna hai (stretch, shrink, ya flip).

Har case cover karo taaki kabhi surprise na ho:

$c > 1$ : lamba, same direction.
$0 < c < 1$ : chhota, same direction.
$c = 0$ : origin tak shrink ho jaata hai (zero vector, agla section).
$c < 0$ : opposite side mein flip ho jaata hai, length original ki $∣ c ∣$ guna.

3. Woh special arrow jo kahin nahi jaata: $0$

4. Arrows ko combine karna: sum $\sum$ aur ek linear combination

Parent likhta hai $\sum_{i} c_{i} v_{i}$ . Aao is symbol ko poori tarah samajhte hain.

Ise zor se padho: "arrow $v_{1}$ ko $c_{1}$ se scale karo, $v_{2}$ ko $c_{2}$ se, …, phir saare scaled arrows ko tip-to-tail jodo."

Counter $i = 1$ se shuru hota hai aur $i = n$ par rukta hai; woh top number $n$ batata hai ki set mein kitne arrows hain — jo dimension ban jaata hai. "Reach everything" side ke liye dekho Span and Spanning Sets aur "no redundancy" side ke liye dekho Linear Independence.

5. Container: vector space $V$ aur ek set $B$

Symbol $R^{2}$ padha jaata hai "R-two": 2 real numbers ki lists. $R^{3}$ : 3 ki lists. Superscript batata hai har vector mein kitne components hain. Kisi bade space ke andar rehne wale chhote spaces ke liye dekho Subspaces.

6. Coordinates: symbol $[w]_{B}$

Ek baar jab tum basis $B$ fix kar lo, toh $w = c_{1} v_{1} + \dots + c_{n} v_{n}$ mein scalars ko ek naam aur notation milta hai.

Case check: standard basis ${e_{1}, e_{2}}$ ke saath, coordinates components ke barabar hoti hain — isliye standard basis "invisible lagti hai." Kisi bhi tilted basis ke saath woh differ karti hain, jo exactly parent ka $(4, 2) \to (3, 1)$ example hai.

7. Logic symbols padhna: $⟹$ , $⟺$ , $\forall$

Proofs teen logic shorthands par lean karti hain. Inhe apna banao taaki proof English jaisi padhi ja sake.

8. Do verbs: span aur independent (simple shabdon mein)

Ab tumhare paas wo saare symbols hain jo basis ki do conditions simple shabdon mein state karne ke liye chahiye.

Basis ka size — har $\sum$ mein top number $n$ — dimension hai, jise Dimension mein explore kiya gaya hai. Aur basis arrows ko ek transformation se feed karna Matrix of a Linear Map banata hai.

Prerequisite map

Equipment checklist

Khud test karo — sirf tab reveal karo jab tum zor se jawab de chuke ho.

Ek "vector" ki minimal job description kya hai?

Tum do ko add kar sako aur ek ko kisi number se scale kar sako.

Scalar

c

vector ke saath kya karta hai, aur agar

c < 0

c = 0

ho toh kya hota hai?

Yeh arrow ko stretch/shrink karta hai;

c < 0

uski direction flip karta hai,

c = 0

use origin par collapse kar deta hai.

0

ki picture kya hai aur poora topic uski parwah kyun karta hai?

Origin par ek dot; yeh test point hai — kitne tareekon se tum wahan pahunchte ho yeh redundancy reveal karta hai.

\sum_{i = 1}^{3} c_{i} v_{i}

ko poori tarah expand karo.

c_{1} v_{1} + c_{2} v_{2} + c_{3} v_{3}

Linear combination kya karne ki permission hai — aur kya NAHI?

Sirf har vector ko scale karo aur add karo; vectors ko aapas mein multiply ya square kabhi nahi.

{v_{1}, v_{2}}

aur

(v_{1}, v_{2})

mein kya fark hai?

Curly braces = poore vectors ka set; round brackets = ek vector ke andar ke number-components.

R^{3}

mein superscript tumhe kya batata hai?

Har vector mein 3 real-number components hote hain.

c_{i} = 0 \forall i

aur

A ⟺ B

ko simple English mein padho.

c_{i}

har index

i

ke liye zero hai"; "

A

tab sach hai exactly jab

B

sach hai."

[w]_{B}

mein subscript

B

kyun hota hai?

Wohi arrow alag basis ke under alag coordinate list pata hai;

B

batata hai kaunsa use hua.

Spanning aur independence dono ko ek-ek simple sentence mein state karo.

Spanning: har vector koi na koi combination hai (existence). Independence: sirf all-zero combination

0

deti hai (uniqueness).

Connections

Linear Independence — "no redundancy" verb jo yahan words mein build hua.
Span and Spanning Sets — "reach everything" verb jo yahan words mein build hua.
Dimension — basis mein arrows ki sankhya $n$ .
Coordinate Vectors and Change of Basis — jahan $[w]_{B}$ centre stage leta hai.
Subspaces — chhote vector spaces jo ek bade $V$ ke andar rehte hain.
Matrix of a Linear Map — basis vectors ko ek map se feed karke banaya jaata hai.
Basis — definition, uniqueness of representation — woh parent jiske liye yeh page tumhe taiyar karta hai.