2.6.4 · D3Scalar multiplication
Yeh page scalar multiplication ke liye "sab kuch ek jagah" wali page hai. Parent note ne rule build kiya tha: k A A har entry ko number k har tarah ke input ke against drill karenge — positive, negative, zero, fractions, zero matrix, ek word problem, aur ek exam trap.
Shuru karne se pehle, ek word jo hum zyada use karte hain: scalar bas ek akela ordinary number hota hai (jaise 3 − 2 2 1 vectors and scalars . Yeh ek matrix nahi hai. Neeche har example mein ek matrix (numbers ki grid, introduction to matrices ) ko ek aisa scalar se multiply kiya gaya hai.
Har scalar-multiplication problem jo tumhe di ja sakti hai, in boxes mein se kisi ek mein aati hai. Hum sabhi ko cover karenge.
Cell
Scenario
Isme tricky kya hai
Example
A
Positive whole scalar, k > 1
Baseline — har entry ko stretch karo
Ex 1
B
Fractional scalar, 0 < k < 1
Shrink; fractions ka dhyan raho
Ex 2
C
Negative scalar, k < 0
Har sign flip ho jaata hai
Ex 3
D
The scalar k = 0
Zero matrix O
Ex 4
E
The scalar k = 1
Kuch nahi badlta (identity check)
Ex 4
F
Combined: k 1 A + k 2 B
Dono baar scale karo, phir add karo
Ex 5
G
Geometric meaning (columns as arrows)
Scaling = picture ko stretch/reflect karna
Ex 6
H
Real-world word problem (units!)
Words → matrix → scale mein translate karo
Ex 7
I
Exam twist: solve for k X
Rule ko ulta chalao
Ex 8
Chalo cell by cell chalte hain.
Worked example Example 1 —k = 4 A = [ 2 5 − 3 0 ] 4 A
Forecast: aage padhne se pehle — har number 4 2 × 2
Scalar ko grid ke aage likho.
4 A = 4 [ 2 5 − 3 0 ] Yeh step kyun? Definition kehti hai k A k 4
Entry-by-entry multiply karo.
= [ 4 ⋅ 2 4 ⋅ 5 4 ⋅ ( − 3 ) 4 ⋅ 0 ] = [ 8 20 − 12 0 ] Yeh step kyun? Koi bhi entry doosri se interact nahi karti (jaise matrix multiplication mein hoti hai); har ek ko apne aap scale kiya jaata hai.
Verify: 4 ⋅ 0 = 0 A 2 × 2 4 A 2 × 2
Worked example Example 2 —k = 3 1 E = [ 6 12 − 9 15 ] 3 1 E
Forecast: 3 1 3 3
3 1 3 1 E = [ 3 6 3 12 3 − 9 3 15 ] Yeh step kyun? Fraction bilkul legal scalar hai; entry-wise rule ko koi fark nahi padta ki k Har fraction simplify karo.
= [ 2 4 − 3 5 ] Yeh step kyun? Har division ek akele number par independent arithmetic hai.
Verify: 3 3 [ 2 4 − 3 5 ] = [ 6 12 − 9 15 ] = E 3 1 3
Worked example Example 3 —k = − 1 B = [ 1 4 2 5 3 6 ] − B
Forecast: − B ( − 1 ) B sign ka kya hoga? Scroll karne se pehle guess karo.
− B − B = ( − 1 ) [ 1 4 2 5 3 6 ] Yeh step kyun? Matrix ke aage bare minus sign ka matlab hi − 1 Har entry ko − 1 = [ − 1 − 4 − 2 − 5 − 3 − 6 ] Yeh step kyun? Kisi number ko − 1
Verify: B + ( − B ) matrix addition ): [ 1 + ( − 1 ) ⋯ ] = [ 0 0 0 0 0 0 ] 0
Worked example Example 4 —0 ⋅ A 1 ⋅ A A = [ 2 5 − 3 0 ] 0 ⋅ A 1 ⋅ A
Forecast: inme se ek matrix ko erase kar dega; doosra use bilkul waise hi chhod dega. Kaunsa kaunsa hai?
Cell D: har entry ko 0 0 ⋅ A = [ 0 ⋅ 2 0 ⋅ 5 0 ⋅ ( − 3 ) 0 ⋅ 0 ] = [ 0 0 0 0 ] = O Yeh step kyun? Zero times koi bhi number zero hota hai, toh har box collapse ho jaata hai — result hai zero matrix O A 0 Cell E: har entry ko 1 1 ⋅ A = [ 1 ⋅ 2 1 ⋅ 5 1 ⋅ ( − 3 ) 1 ⋅ 0 ] = [ 2 5 − 3 0 ] = A Yeh step kyun? One times koi bhi number wahi number hota hai, toh kuch nahi hilta — 1
Verify: dono cases mein size unchanged rehti hai (2 × 2 2 × 2 0 ⋅ A = O
0 ⋅ A 0 matrix hai, zeros ki poori grid jiska A 2 × 2
Worked example Example 5 —3 C + 2 D C = [ 1 0 0 1 ] D = [ 2 4 3 5 ] 3 C + 2 D
Forecast: do scalings hoti hain pehle , phir koi addition hoti hai. Pehle 3 C 3
C 3 3 C = [ 3 0 0 3 ] Yeh step kyun? Rule D C D 2 2 D = [ 4 8 6 10 ] Yeh step kyun? Same rule, alag scalar aur matrix.Dono scaled matrices ko entry-by-entry add karo.
3 C + 2 D = [ 3 + 4 0 + 8 0 + 6 3 + 10 ] = [ 7 8 6 13 ] Yeh step kyun? Matrices ka addition bhi entry-wise hota hai, aur dono 2 × 2
Verify: top-left hai 3 ( 1 ) + 2 ( 2 ) = 7 3 ( 1 ) + 2 ( 5 ) = 13 linear-combination pattern hai.
Yahan 2 × 2 columns ko arrows ki tarah padha jaata hai jo origin se shuru hote hain. Matrix ko scale karna dono arrows ko scale karta hai. Yeh linear transformations ka seed hai.
Worked example Example 6 — arrows ko stretch aur reflect karo (Cell G)Maano M = [ 2 1 − 1 1 ] u = [ 2 1 ] v = [ − 1 1 ] 2 M ( − 1 ) M
Forecast: doubling se har arrow ki length double honi chahiye, same direction mein. − 1
2 2 M = [ 4 2 − 2 2 ] Yeh step kyun? Har column ke coordinates double ho jaate hain, toh har arrow ki length double ho jaati hai jabki uska angle fixed rehta hai.
Blue original arrow u yellow 2 u
− 1 ( − 1 ) M = [ − 2 − 1 1 − 1 ] Yeh step kyun? − 1
Red arrow − u u 18 0 ∘
Verify: u 2 2 + 1 2 = 5 2 u = 4 2 + 2 2 = 20 = 2 5 − u = ( − 2 ) 2 + ( − 1 ) 2 = 5
Worked example Example 7 — price list double karna (Cell H)Ek shop weekly sales ko matrix S S = [ 30 50 20 15 ] prices double kar deta hai (abhi ke liye units same rehte hain). Lekin prices sirf column 2 mein hain. Koi naive hokar 2 S
Forecast: kya poori matrix scale hoti hai, ya sirf ek column? Socho ki "prices double karna" actually kya touch karta hai.
Pehle dekho ki 2 S kya dega .
2 S = [ 60 100 40 30 ] Yeh step kyun? Scalar multiplication all-or-nothing hai — yeh sirf ek column touch nahi kar sakta. Toh 2 S units sold bhi double kar deta hai.
Sahi model: sirf column 2 price hai. Ek akele column ko scale karna S nahi hai; us column ko alag se scale karo.
S new = [ 30 50 40 30 ] Yeh step kyun? Jo physical quantity double ho rahi hai (price, ₹ mein) woh sirf column 2 mein hai, toh sirf wahi entries badlti hain.
Verify: units column unchanged: 30 , 50 20 → 40 15 → 30 Lesson: scalar multiplication sab kuch scale karta hai; ise tabhi use karo jab poori grid same tarah ki quantity ho jo same factor se scale ho rahi ho.
k A k har entry ko equally hit karta hai. Agar tumhara real-world factor sirf kuch rows/columns par apply hota hai (ya alag-alag parts par alag factors), toh poori matrix ki scalar multiplication galat tool hai.
Worked example Example 8 — unknown ke liye solve karo (Cell I)Scalar k X k [ 3 − 6 ] = [ 12 − 24 ] , 2 X = [ 8 6 − 4 0 ] .
Forecast: part 1 mein tumhe before aur after pata hai, toh k 2
k k k ⋅ 3 = 12 ⟹ k = 3 12 = 4 Yeh step kyun? Kyunki k ek hi number hai jo har jagah apply hota hai, ek equation use determine kar deti hai.Doosri entry ke saath consistency check karo.
4 ⋅ ( − 6 ) = − 24 ✓ Yeh step kyun? Agar dono entries alag k
X 2 1 2 X = 2 1 [ 8 6 − 4 0 ] = [ 4 3 − 2 0 ] Yeh step kyun? 2 1 ⋅ 2 = 1 1 ⋅ X = X 2 1 X
Verify: wapas plug karo — 2 [ 4 3 − 2 0 ] = [ 8 6 − 4 0 ] 4 [ 3 − 6 ] = [ 12 − 24 ]
Recall Kaun se scenarios matrix ko collapse ya preserve karte hain?0 ⋅ A 1 ⋅ A A 0 ⋅ A O 1 ⋅ A A
Recall Negative scalar ka matrix ke columns par geometric effect kya hota hai?Har column-arrow ko origin ke through reflect karo (same length, opposite direction).
Effect of multiplying by − 1 18 0 ∘
Mnemonic "Ek scalar, poori grid, koi exception nahi"Agar factor har entry par equally apply nahi hota, toh yeh matrix ki scalar multiplication nahi hai.