Basis — definition, uniqueness of representation

4.5.17 · Maths › Linear Algebra (Full)

1. Shuru se: humein kaun se do properties chahiye?

Hum chahte hain ki ek set $B = {v_{1}, \dots, v_{n}}$ jo vector space $V$ ke andar ho, coordinate axes ki tarah kaam kare.

Hum $B$ se koi bhi vector build kar sakein → yeh spanning hai.
Hamein koi redundancy na ho (koi axis doosron se express na ho sake) → yeh linear independence hai.

2. Central theorem (DERIVE karo)

" $\Rightarrow$ " ka proof: basis uniqueness deta hai.

Existence. Spanning literally kehta hai ki koi $w = \sum c_{i} v_{i}$ exist karta hai. ✔

Uniqueness. Maano $w$ ke do representations hain: $w = \sum_{i} c_{i} v_{i} aur w = \sum_{i} d_{i} v_{i} .$ Inhe subtract karo (valid hai — same vector minus itself): $0 = w - w = \sum_{i} (c_{i} - d_{i}) v_{i} .$ Yeh step kyun? Kyunki "do barabar sums" ko "kuch $= 0$ " mein badalna humein independence ko invoke karne deta hai, jo zero ke barabar combinations ke baare mein ek statement hai.

Linear independence se, $\sum_{i} (c_{i} - d_{i}) v_{i} = 0$ ka ek hi tarika hai ki har coefficient zero ho: $c_{i} - d_{i} = 0 \Rightarrow c_{i} = d_{i} \forall i .$ Toh dono representations ek hi the. Uniqueness proved. ∎

" $\Leftarrow$ " ka proof: uniqueness basis deta hai.

Sabke liye existence = spanning. ✔
$w = 0$ par uniqueness apply karo: ek representation hai $0 = \sum 0 \cdot v_{i}$ . Uniqueness kehti hai yahi sirf ek hai, toh $\sum c_{i} v_{i} = 0 \Rightarrow c_{i} = 0$ → yahi exactly independence hai. ✔ ∎

Figure — Basis — definition, uniqueness of representation

3. Worked examples

4. Common mistakes (Steel-manned)

5. The 80/20 core

6. Flashcards

Basis define karne ke liye kaun si do conditions chahiye?

Linear independence AUR

V

ka spanning.

Kaun si basis property representation ki existence guarantee karti hai?

Spanning.

Kaun si basis property representation ki uniqueness guarantee karti hai?

Linear independence.

Uniqueness theorem state karo.

B

ek basis hai

⟺

har

w \in V

uniquely

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

hai.

Uniqueness proof mein, do reps assume karne ke baad kaun sa trick use hota hai?

Unhe subtract karo taaki

\sum (c_{i} - d_{i}) v_{i} = 0

mile, phir independence apply karo.

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

mein

c_{i}

ko kya kehte hain?

Basis

B

ke relative

w

ke coordinates.

{(1, 1), (1, - 1)}

basis mein

(4, 2)

ke coordinates kya hain?

(3, 1)

{e_{1}, e_{2}, (1, 1)}

kyun

R^{2}

ka basis nahi hai?

Yeh span karta hai lekin dependent hai, toh coordinates unique nahi hote.

{e_{1}, e_{2}}

kyun

R^{3}

ka basis nahi hai?

Independent hai lekin span nahi karta;

(0, 0, 1)

tak nahi pahuncha ja sakta.

w = 0

ke liye uniqueness independence kaise recover karta hai?

0

ka sirf ek rep hai jo all-zero coefficients hai, aur yahi independence ki definition hai.

Kisi bhi basis ka size kiske barabar hota hai?

dim V

Recall Feynman: 12-saal ke bachche ko samjhao

Socho ki ek grid par "East ke steps" aur "North ke steps" use karke raasta bata rahe ho. Yahi tumhara basis hai. Spanning ka matlab hai ki tum kisi bhi jagah pahunch sako. Independence ka matlab hai ki North secretly East jaisa nahi hai — yeh genuinely alag directions hain. Isi wajah se, har jagah ka ek honest jawab hota hai jaise "3 East, 1 North." Agar main tumhe ek teesra bekar direction deta (jaise "Northeast"), toh achaanak usi jagah ko kai tarakon se describe kiya ja sakta tha — confusing! Ek basis directions ka sabse clean possible set hai: har jagah pahunchne ke liye kaafi, lekin koi bacha hua nahi, toh har jagah ka exactly ek naam ho.

Connections

Linear Independence — uniqueness ka aadha hissa deta hai.
Span and Spanning Sets — existence ka aadha hissa deta hai.
Dimension — har basis ka invariant size.
Coordinate Vectors and Change of Basis — coordinates $[w]_{B}$ chosen $B$ par depend karte hain.
Subspaces — bases humein subspace dimension measure karne deti hain.
Matrix of a Linear Map — basis vectors ko map se guzaar kar banaya jaata hai.