Change of basis matrix: Foundations — Hinglish deep dive

4.5.20 · D1 · Maths › Linear Algebra (Full) › Change of basis matrix

Uss machine ko build karne se pehle, tumhe raw ingredients mein fluent hona chahiye. Neeche har symbol aur idea hai jis par parent note depend karta hai, build-order mein. Koi cheez use nahi hoti jab tak woh draw na ho.

1. Vector kya hai? (woh arrow jo kabhi hilta nahi)

Letter $v$ (ya $b_{1}$ , $w$ , ...) bas ek aise arrow ka naam hai.

Yeh topic ko kyun chahiye: "change of basis" ka poora point yahi hai ki yeh arrow usi jagah rehta hai jabki uske numbers badal jaate hain. Agar tum bhool gaye ki arrow fixed hai, toh har sign aur inverse galat ho jaayega. Figure mein dekho: amber arrow dono panels mein same hai.

2. Plane $R^{n}$ aur origin $O$

$O$ = origin, har arrow ka shared starting point (jahan saare axes cross karte hain).
$n$ = dimension, un directions ki sankhya jo tumhe chahiye. $R^{2}$ mein, $n = 2$ .

Yeh topic ko kyun chahiye: coordinates ki sankhya $n$ ke barabar hai, aur change of basis matrix ek $n \times n$ square hai. Basis and Dimension dekho ki $n$ kahan se aata hai.

3. Arrows ko scale karna aur add karna

Do operations arrows ko ek language mein badal deti hain.

Yeh topic ko kyun chahiye: har coordinate expansion "kuch reference arrows ko scale karo, phir unhe add karo" hai. Formula $v = c_{1} b_{1} + c_{2} b_{2}$ kuch nahi balki Step 1 (scale) ke baad Step 2 (add) hai, baar baar kiya gaya.

4. Linear combination

Sigma shorthand padhna. Bada Greek "S" ( $Σ$ , sigma, sum ke liye) us poori line ko pack karta hai: $\sum_{j = 1}^{n} c_{j} b_{j} = c_{1} b_{1} + c_{2} b_{2} + \dots + c_{n} b_{n} .$ $\sum$ ke neeche letter $j$ ek counter hai jo $1, 2, \dots, n$ walk karta hai; har value ke liye woh $j$ -waan scalar $c_{j}$ aur $j$ -waan arrow $b_{j}$ leta hai, aur tum saare pieces add karte ho. Yeh sirf lambe sum likhne ka compact tarika hai — kuch naya nahi.

Yeh topic ko kyun chahiye: parent ke Steps 1–3 $\sum$ signs mein dabbe hue hain. Agar $\sum$ ek mystery hai, toh derivation unreadable hai.

5. Linear independence & span → basis

Yeh topic ko kyun chahiye: ek basis hi woh cheez hai jo guarantee karti hai ki har vector ka exactly ek coordinate description ho — na zero, na bahut saare. Independence duplicates ko khatam karta hai; spanning coverage guarantee karta hai. Full detail Basis and Dimension mein hai.

6. Coordinates aur column $[v]_{B}$

Notation ko piece by piece decode karo:

$[v]_{B}$ — "arrow $v$ ka coordinate column, basis $B$ se measure kiya gaya." Chota subscript $B$ batata hai ki tumne kaun sa ruler set use kiya.
Tall bracket $c_{1} ⋮ c_{n}$ — ek column vector: numbers upar-se-neeche likhe gaye. Teen dots $⋮$ ka matlab hai "same tarike se continue karo."

$e_{j}$ (parent mein use hota hai) ek special coordinate column hai: sab zeros siwaaye slot $j$ mein ek single $1$ ke. Socho isko "direction $j$ ki taraf ek poora step, doosron ki taraf kuch nahi."

7. Matrix × column: translator ka engine

Yahan $p_{ij}$ ka matlab hai "row $i$ , column $j$ mein number" — row pehle, column baad mein, hamesha.

Yeh topic ko kyun chahiye: change of basis matrix act karta hai matrix–vector multiplication se. Derivation ka final Step 4 literally "sum ko $P x$ ke roop mein pehchano" hai.

8. Inverse matrix $P^{- 1}$

Ek $2 \times 2$ block ke liye ek hand formula hai: $[a c b d]^{- 1} = \frac{1}{a d - b c} [d - c - b a],$ valid jab number $a d - b c$ (determinant) nonzero ho.

Yeh topic ko kyun chahiye: "translate karo phir translate back karo" identity hai, isliye reverse dictionary $P_{B \leftarrow C} = (P_{C \leftarrow B})^{- 1}$ hai. Ek basis hamesha independent hoti hai, isliye uski matrix mein hamesha nonzero determinant hota hai aur inverse hamesha exist karta hai — woh guarantee Invertible Matrices hai.

ek fixed arrow (length + direction) origin se, kisi bhi numbers se independent.

R^{n}

ka matlab hai

n

real numbers ki saari ordered lists — ek

n

-dimensional space;

n

independent directions count karta hai.

Ek scalar hota hai

ek ordinary real number jo ek arrow ko stretch (aur possibly flip) karta hai.

Do arrows add karne ke liye

unhe tip-to-tail rakho; sum pehli tail se aakhri tip tak jaata hai.

Ek linear combination hoti hai

c_{1} b_{1} + \dots + c_{n} b_{n}

— har arrow ko scale karo, phir add karo.

\sum_{j = 1}^{n} c_{j} b_{j}

shorthand hai

poore sum

c_{1} b_{1} + c_{2} b_{2} + \dots + c_{n} b_{n}

ke liye;

j

ek counter hai.

Ek basis hoti hai

arrows ka ek aisa set jo linearly independent aur spanning dono ho (minimal + complete).

Basis unique coordinates kyun deta hai

independence same vector ke do alag expansions ko rokta hai.

[v]_{B}

ka matlab hai

arrow

v

ko basis

B

use karke describe karne wale scalars ka column.

e_{j}

hota hai

coordinate column jisme slot

j

mein

1

aur baaki jagah

0

ho.

P x

columns se compute karna hai

P

ke columns ki linear combination

x

ki entries se weighted;

P e_{j}

= column

j

P^{- 1}

satisfy karta hai

P^{- 1} P = P P^{- 1} = I

; yeh

P

ko undo karta hai.

2 \times 2

inverse formula

\frac{1}{a d - b c} [d - c - b a]

, valid jab

a d - b c \neq = 0

ho.

Passive vs active

passive = numbers change, arrow fixed (yeh topic); active = arrow actually move karta hai.

Connections

Basis and Dimension — jahan se number $n$ aur "independent + spanning" aate hain.
Coordinate Vectors — columns $[v]_{B}$ jinhe hum manipulate karte hain.
Invertible Matrices — guarantee karta hai ki $P^{- 1}$ ek basis ke liye hamesha exist karta hai.
Linear Transformations — is passive story ka active cousin.
Similar Matrices — actual maps par $P^{- 1} (\cdot) P$ reuse karta hai.
Eigenvectors and Diagonalization — ek special basis mein change karna.
Change of basis matrix — woh parent jiske liye yeh page tumhe tayaar karta hai.

Foundations — Change of basis matrix