Foundations — Coordinate vectors — change of basis

4.5.19 · D1 · Maths › Linear Algebra (Full) › Coordinate vectors — change of basis

Is page par assume kiya gaya hai ki parent note ki notation ke baare mein tumhe kuch nahi pata. Hum har symbol ek-ek karke build karte hain, ek doosre ke upar. Upar se neeche padho.

1. Arrow: "vector" actually kya hota hai

Figure dekho. Lal arrow origin (dot) se shuru hota hai aur kahin point karta hai. Woh hi $\mathbf{v}$ hai. Dhyan do: humne abhi tak koi numbers nahi likhe. Arrow measure karne se pehle exist karta hai — yeh is page ka sabse important idea hai.

2. Numbers, aur $\left[\begin{array}{c}\end{array}\right]$ column

Tall bracket bas ek container hai. Yeh scalars ki ek list ko ek fixed order mein rakhta hai. Order matter karta hai: $\left[\begin{array}{c}4\\ 2\end{array}\right]$ aur $\left[\begin{array}{c}2\\ 4\end{array}\right]$ alag-alag arrows hain.

3. Measuring-arrows: basis ka idea

Yeh woh subtle part hai jis par poora topic depend karta hai. "4 steps right, 2 steps upar" secretly assume karta hai ki ek step ka kya matlab hai. Kaun si direction mein step? Kitna?

Figure dekho: same target point tak do alag sets of measuring-arrows se pahoncha ja sakta hai. Left mein ordinary right/up arrows use hote hain; right mein do slanted arrows. Same destination, alag instructions.

4. Subscripts, "$\dots$" dots, aur $n$

• Subscript ${\mathbf{b}}_j$: chhota $j$ ek name tag hai, power nahi. ${\mathbf{b}}_1,{\mathbf{b}}_2,{\mathbf{b}}_3$ teen alag arrows hain.
• $n$: ek stand-in hai "basis mein jitne bhi arrows hoon" ke liye. Plane mein $n=2$; 3D space mein $n=3$. Yeh count dimension hai (Basis and dimension).
• Dots $\dots$: shorthand hai "obvious pattern mein chalte raho." $\{{\mathbf{b}}_1,\dots ,{\mathbf{b}}_n\}$ ka matlab hai "${\mathbf{b}}_1$, ${\mathbf{b}}_2$, aur aise hi ${\mathbf{b}}_n$ tak."

5. Measuring-arrows se arrow banana: $\sum$ aur linear combination

Picture dikhata hai $\mathbf{v}$ ko ${\mathbf{b}}_1$ ki $c_1$ copies end-to-end rakhke, phir ${\mathbf{b}}_2$ ki $c_2$ copies se banana.

6. Coordinate vector $[\mathbf{v}{]}_{\mathcal{B}}$ aur yeh UNIQUE kyun hai

7. Basis arrows ko matrix $B$ mein stack karna

8. $B$ ko undo karna: inverse $B^{-1}$

Agar $B$ $\mathcal{B}$-numbers ko ordinary numbers mein convert karta hai, toh hume ek machine chahiye jo ulta jaaye.

In do machines ke saath, parent note ka punchline clearly padhta hai: $\underset{\mathcal{C}\leftarrow \mathcal{B}}{P}=C^{-1}B\text{— pehle }B\text{ lagaao (standard par), phir }{C}^{-1}\text{ (standard se }\mathcal{C}\text{ par).}$

Chhota arrow $\mathcal{C}\leftarrow \mathcal{B}$ $P$ ke neeche right-to-left padhta hai: "input $\mathcal{B}$-numbers hain, output $\mathcal{C}$-numbers hain." Jab tum ise Similar matrices and diagonalization se connect karoge, toh wohi $C^{-1}(\cdot )B$ pattern phir se dekhoge.

Prerequisite map

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Parent topic: Coordinate vectors — change of basis.