Foundations — Scalar multiplication

2.6.4 · D1 · Maths › Matrices & Determinants — Introduction › Scalar multiplication

Ye page ground floor hai. Parent note Scalar multiplication mein bahut saari notation ek saath aayi hain — $a_{ij}$, $m\times n$, $\mathbb{R}$, $[\cdot ]$, zero matrix $O$. Hum unme se ek bhi use nahi karenge jab tak wo scratch se build na ho jaaye. Upar se neeche padho; har block sirf upar waale blocks ke ideas use karta hai.

1. Ek number (ek "scalar")

Topic ko "scalar" ka alag word kyun chahiye? Kyunki hum abhi numbers ki grids se milne waale hain. Hume ek word chahiye "ek akela single number" ($k$) ke liye aur ek alag word "poori grid" ($A$) ke liye — taaki hum kabhi confuse na hon. Wo akela single number hi scalar hai.

Scalar ki picture ek number line par point hoti hai: $0$ se uski doori uska size hai, aur uski side uska sign batati hai.

Tum scalars ko 2.2.04-Vectors-and-scalars se already jaante ho — wahan scalar ko vector (kuch direction waala) se contrast kiya jaata hai. Yahan hum ise grid se contrast kar rahe hain.

2. Ek number ko doosre se multiply karna = stretching

Yahi poore topic ka beej hai. Ek grid ko scale karne se pehle, ek akele number ko scale karna bilkul clear hona chahiye.

• $k=2$: number $0$ se do guna door ho jaata hai (stretch).
• $k=\frac{1}{2}$: $0$ se aadha door (shrink).
• $k=-1$: same doori, lekin opposite side ($0$ ke through flip).
• $k=0$: bilkul $0$ par ja girta hai, chahe pehle kahan bhi tha.
• $k=1$: bilkul nahi hilta.

Upar ke paanch behaviours noto karo — stretch, shrink, flip, collapse-to-zero, stay-put. Ye sab baad mein poori grid ke saath hoga, cell by cell. Kuch naya nahi add hota; yahi move hai jo kai baar dohraaya jaata hai.

3. Numbers ki ek row aur ek column

Poori grid se pehle, uski do building strips se milo.

Picture: row mein numbers kaandhe se kaandha milakar khade hote hain; column mein numbers sir se pair tak stack hote hain. Topic ko dono words chahiye kyunki hum count karenge ki ek grid mein kitni rows aur kitne columns hain — woh count uski shape hai.

4. Grid: ek matrix, aur bracket $[]$

Bracket kyun? Taaki aankhon ko turant ek hi cheez dikhe (ek grid $A$), na ki chaar alag numbers. Yahi wo "container" hai jiske baare mein parent note baat karta hai. Poora detail 2.6.01-Introduction-to-matrices mein hai.

Picture: ek filing cabinet — rows aur columns ke fixed pigeon-holes, har hole mein ek number.

5. Ek jagah ka naam: $a_{ij}$

Ab notation ka woh sabse tricky piece jo parent note mein sabse zyada use hota hai.

Ise zabanaan padho: "$a$, row $i$, column $j$." To $A=\left[\begin{array}{cc}2& -3\\ 5& 0\end{array}\right]$ mein:

• $a_{11}=2$ (row 1, column 1),
• $a_{12}=-3$ (row 1, column 2),
• $a_{21}=5$ (row 2, column 1),
• $a_{22}=0$ (row 2, column 2).

Topic ko $a_{ij}$ kyun chahiye? Kyunki scalar multiplication ek saath har hole par hoti hai, aur $a_{ij}$ woh akela symbol hai jo keh sakta hai "jo bhi number hole $(i,j)$ mein baitha hai use lo" — bina sab kuch likhe. Yeh ek placeholder hai jo saari entries ko ek saath represent karta hai.

6. Shape: $m\times n$

Matrix $A=\left[\begin{array}{cc}2& -3\\ 5& 0\end{array}\right]$ mein 2 rows aur 2 columns hain, isliye yeh $2\times 2$ matrix hai. Grid $\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]$ $2\times 3$ hai (2 rows, 3 columns).

Picture: $m\times n$ filing cabinet ka frame size hai. Scalar multiplication holes ke andar ki cheez badlegi lekin frame kabhi nahi — isliye hume frame ka naam chahiye taaki keh sakein "frame same rehti hai."

7. Do shorthand symbols: $[a_{ij}{]}_{m\times n}$ aur $\mathbb{R}$

$\mathbb{R}$ kyun introduce karein? Taaki topic ek sentence ki jagah teen symbols mein keh sake "koi bhi real scalar allowed hai." $\in$ symbol ka matlab bas "is mein shamil hai / iska member hai."

8. Zero matrix $O$

Uski picture: woh filing cabinet jisme har drawer khaali hai ($0$ rakh ke). Hume iska zaroorat isliye hai kyunki jab scalar $k=0$ hota hai, to har hole $0$ par collapse ho jaata hai (§2 ka "collapse-to-zero" move yaad karo), to poori grid $O$ ban jaati hai. Parent note isse $0\cdot A=O$ likhta hai.

9. Sab kuch jodna — $kA=[k{a}_{ij}{]}_{m\times n}$ padhna

Ab parent ki headline formula mein kuch bhi aisa nahi jo tumne build na kiya ho:

$kA=[k{a}_{ij}{]}_{m\times n}$

Dheere padho, symbol by symbol, upar ke sections use karke:

• $k$ — ek akela scalar, line par ek point (§1), ek stretch factor (§2).
• $A$ — grid (§4).
• $a_{ij}$ — jo bhi number row $i$, column $j$ mein baitha hai (§5).
• $k{a}_{ij}$ — us hole ka number $k$ se stretched (§2), ek hole mein kiya gaya single-number move.
• $[\cdots {]}_{m\times n}$ — un saare stretched holes ko same frame size ki grid mein wapis ikatta karo (§6, §7).

To formula poori English mein kehta hai: "$kA$ banane ke liye, grid ke har hole par jao aur uske number ko us number times $k$ se replace karo, frame unchanged rakhte hue." Yahi poora topic hai.

Prerequisite map

Yeh foundation aage feed karta hai: ek baar jab tum grid ko scale kar sako, to ise matrix addition ke saath combine kar sakte ho, mushkil matrix multiplication se mil sakte ho, shared rules ko properties of matrix operations mein dekh sakte ho, aur baad mein grids ko linear transformations ki tarah act karte dekh sakte ho.

Equipment checklist

Right side cover karo aur parent topic par jaane se pehle har ek ka jawab do.