Properties — row operations, multiplicativity: Foundations

4.5.22 · D1 · Maths › Linear Algebra (Full) › Properties — row operations, multiplicativity

Ye page assume karta hai ki aapne kuch bhi nahi dekha. Hum har wo symbol name karte hain jis par parent note depend karta hai, uske peeche ki picture draw karte hain, aur batate hain ki topic uske bina kyun nahi chal sakta. Upar se neeche padho — har idea agli ke liye ek rung hai.

1. Number, scalar, aur letter $k$

Picture: ek scalar ek stretch factor hai. Kisi cheez ko $k = 2$ se multiply karna use do guna lamba kar deta hai; $k = \frac{1}{2}$ use aadha shrink karta hai; $k = - 1$ uski direction flip karta hai.

Topic ko ye kyun chahiye: rule "ek row ko $k$ se scale karo" aur corollary $det (k A) = k^{n} det A$ poori tarah is baat ke baare mein hain ki ek scalar length aur area ke saath kya karta hai. Ek row ko stretch karne ki baat karne se pehle, hume stretch number ka idea chahiye.

2. Vector aur "a row"

Picture: figure s01 mein row $(2, 3)$ ko ek arrow ki tarah draw kiya gaya hai jo point $(2, 3)$ tak right aur upar jaata hai. Numbers batate hain kitna right aur kitna upar.

Figure — Properties — row operations, multiplicativity

Topic ko ye kyun chahiye: poori derivation determinant ko uske rows ka ek function likhti hai, $D (R_{1}, \dots, R_{n})$ . "Do rows swap karo" ya "ek row ka multiple doosri mein add karo" samajhne ke liye, pehle aapko har row ko ek aisa arrow dekhna hoga jo aap move kar sako.

3. Matrix — rows ka ek stack, aur ek machine

Picture: figure s02 dono readings ek saath side by side dikhata hai. Left par, $A$ do arrows hain (uski rows). Right par, $A$ ek machine ki tarah kaam karta hai: chhota unit square andar daalo, aur wo stretch aur tilt hokar ek parallelogram mein bahar aata hai.

Topic ko ye kyun chahiye: parent note ke saare objects matrices hain, capital letters se named: $A$ aur $B$ general machines hain, $I$ do-nothing machine hai (§7), ek $E$ ek aisi machine hai jo ek row operation perform karta hai (§8b), aur ek $U$ ek triangular machine hai (§10). Har ek apne section mein properly introduce kiya gaya hai neeche. "Row operations" stack reading ko edit karte hain; $A B$ jaise products machine reading use karte hain. Aapko dono pictures ek saath dhyan mein rakhni hogi.

4. Unit square, area, aur orientation

Picture: figure s03. Machine unit square (area $1$ ) ko ek parallelogram mein badal deti hai. Do cheezein ho sakti hain:

parallelogram ka area bada ya chhota ho sakta hai — yahi scaling factor hai;
corners ulti taraf (clockwise instead of counter-clockwise) ghoom sakte hain — yahi flip hai, jo chhote curved arrow ke reverse hone se dikhaya gaya hai.

Topic ko ye kyun chahiye: yahi determinant ki definition hai jis se parent shuru hota hai — "unit cube ke volume ka signed scaling factor". Parallelogram ka size $∣ det A ∣$ hai; sign record karta hai ki space flip hua ya nahi. Row swaps flip karte hain; row scalings resize karte hain; shears corners ko sideways slide karte hain bina area change kiye.

5. Determinant symbol $det$

Picture: figure s03 mein shaded parallelogram; uska signed area hi $det A$ hai.

Topic ko ye kyun chahiye: ye hi topic hai. Baaki sab measure karta hai ki edits par $det$ kaise react karta hai.

5b. Area (2D) se volume (3D aur aage) tak

Picture: figure s05. Space mein teen row-arrows ek tedha box frame karte hain; $det A$ batata hai ki kitne unit cubes us mein fit hote hain, $-$ sign ke saath agar teen arrows left-handed set banate hain right-handed ki jagah (3D version of "mirror-flipped").

Topic ko ye kyun chahiye: har rule jo aap milenge — swap sign flip karta hai, ek row ko $k$ se scale karne par $k$ se multiply hota hai, shear kuch nahi karta, $det (k A) = k^{n} det A$ — sab general $n \times n$ matrices ke liye stated hain. 2D pictures sirf sabse easy-to-draw case hain; jab bhi text "volume" kahe parallelepiped picture karo, aur trust karo ki same verbal rule har dimension mein hold karta hai. Yahi transfer ki wajah hai ki hum rules ko axioms se prove karte hain na ki ek single drawing se.

6. Subscripts, arrows, aur operation notation

Picture: §2 ke arrows ki tarah rows ke baare mein socho. Ek swap do arrows ki positions trade karta hai; ek scale ek arrow ko stretch karta hai; ek replacement ek arrow ki tip ko doosre ki direction mein slide karta hai (ek shear), figure s04 mein dikhaya gaya hai — notice karo ki base wahi rehta hai aur area nahi badlta.

Topic ko ye kyun chahiye: ye teen arrows hi wo teen elementary row operations hain jis ke around poora chapter revolve karta hai.

7. Identity matrix $I$

$I_{2} = (1001), det I = 1.$

Picture: "do-nothing" machine unit square ko unit square hi rehne deti hai — area $1$ , koi flip nahi. Isliye $det I = 1$ hai, normalization axiom.

Topic ko ye kyun chahiye: ye reference point hai. $det I = 1$ poore system ko anchor karta hai, aur $det (A^{- 1}) = 1/ det A$ $A A^{- 1} = I$ se aata hai.

8. Multiplication $A B$ aur inverse $A^{- 1}$

Picture: unit square ko $B$ se pass karo (area $\to det B$ ), phir $A$ se (area phir $det A$ se multiply ho). Total area factor $= det A \cdot det B$ . Ye hai $det (A B) = det A det B$ dekha hua.

Topic ko ye kyun chahiye: multiplicativity — parent ki doosri headline — literally "areas multiply when machines chain" hai.

8b. Elementary matrix $E$

Jaise, $I_{2}$ se start karke aur uski do rows swap karne par swap matrix milta hai $E = (0110), det E = - 1.$

Picture: $E$ ek "one-move machine" hai — ye exactly ek swap, ek scale, ya ek shear karta hai aur kuch nahi. Isliye uska determinant exactly woh factor hai jo wo move apply karta hai: swap ke liye $- 1$ , scale ke liye $k$ , shear ke liye $1$ .

Topic ko ye kyun chahiye: parent ka algebraic proof of $det (A B) = det A det B$ , $A$ ko in one-move machines $E_{1} E_{2} \dots E_{m}$ ki chain ki tarah likhta hai. Proof samajhne se pehle aapko pata hona chahiye ki letter $E$ kya stand karta hai. Dekho Elementary Matrices.

9. Product notation $\prod$ , powers $A^{k}$ , aur exponent $n$

Topic ko ye kyun chahiye: REF formula $det U = \prod_{i} u_{ii}$ aur corollaries $det (A^{k}) = (det A)^{k}$ , $det (k A) = k^{n} det A$ sab is compact notation use karte hain.

10. Triangular matrix aur "pivot"

$U = 200310 13 - 3, det U = 2 \cdot 1 \cdot (- 3) = - 6.$

Intuition Kyun triangular

$det$ = diagonal ka product hai Geometric: pehla row-arrow $x$ -axis ke along $u_{11}$ length ke saath hai (uske baad ke entries base ke liye matter nahi karte). Doosra arrow us axis ke upar $u_{22}$ height tak uthta hai — diagonal ke neeche zeros ka matlab hai ki har naya arrow ek fresh perpendicular height add karta hai aur koi sideways cheez nahi jo box ko collapse kar sake. To tedha box actually "base $\times$ height $\times$ height $\dots$ " hai, yaani $u_{11} \cdot u_{22} \dots u_{nn}$ . Shearing (diagonal ke upar non-zero entries) sirf corners ko sideways slide karta hai aur wo product kabhi nahi badlta. Algebraic: determinant ko pehle column ke along expand karo. Kyunki $u_{11}$ ke neeche har entry $0$ hai, sirf top-left term bachta hai, jo $u_{11}$ times chhote triangular block ka determinant hai. Ye process poori tarah neeche repeat karo aur aap $u_{11} u_{22} \dots u_{nn}$ par pahunch jaate ho.

Topic ko ye kyun chahiye: koi bhi $det$ compute karne ka practical tarika hai matrix ko shear-and-swap karke triangular form mein laana, phir pivot product padhna. Ye Gaussian Elimination & Row Echelon Form ka bridge hai.

11. Teen defining properties (derivation ki vocabulary)

Picture: do equal rows = do identical arrows = ek degenerate parallelogram jisme koi area nahi. Ye ek fact (A) hi hai jo swap par sign-flip ko force karta hai.

Topic ko ye kyun chahiye: parent har rule sirf (M), (A), (N) se derive karta hai. Ye teen words axioms hain — dekho Multilinear Alternating Forms.

Prerequisite map

Equipment checklist

Right side cover karo aur khud ko test karo — aap parent topic ke liye ready hain jab har line instantly aa jaaye.

Scalar

k

geometrically length ke saath kya karta hai?

Use

k

factor se stretch karta hai (agar

k < 0

to direction flip hoti hai).

Ek row, picture mein kya hai?

Origin se un coordinates wale point tak ek arrow.

Matrix ki do readings kya hain?

Row-arrows ka ek stack, aur ek machine jo unit square ko parallelogram mein map karti hai.

det A

geometrically kaun sa number hai?

Us output parallelogram ka signed area (

+

orientation rakhta hai,

-

flip karta hai).

3D mein unit cube kaun si shape ban jaata hai aur

det

kya hai?

Ek tedha box (parallelepiped);

det

uska signed volume hai.

det (a c b d) =

a d - b c

Rows par arrows

\leftrightarrow

aur

\to

ka kya matlab hai?

\leftrightarrow

do rows swap karta hai;

\to

ek row replace karta hai (scale karo, ya doosre ka multiple add karo).

Kaun sa row operation shear hai, aur area ka kya hota hai?

R_{i} \to R_{i} + k R_{j}

; area (aur

det

) unchanged rehta hai.

I

kya hai aur

det I

kya hai?

Do-nothing matrix (diagonal par 1's);

det I = 1

Elementary matrix

E

kya hai?

Identity jis par ek row operation kiya gaya ho;

E A

woh operation

A

par perform karta hai.

A B

ka kya matlab hai aur

det (A B)

kya equal hai?

Pehle

B

karo phir

A

;

det (A B) = det A det B

det (k A) = k^{n} det A

kyun hai, na ki

k det A

Saare

n

rows scale hote hain, to ek factor

k

har row se nikalta hai.

Triangular matrix ka

det

kya hai, aur kyun?

Uske diagonal pivots ka product; column expansion har step par sirf top-left term chodta hai (ya: base

\times

successive heights).

det

ki teen defining properties bolo.

Har row mein Multilinear, Alternating (equal rows

\Rightarrow 0

), Normalized (

det I = 1

Ready? Parent Row Operations & Multiplicativity note mein jo har symbol use hota hai uska ab ek plain meaning aur ek picture hai. Agle stops: Determinant — Definition (cofactor / Leibniz), Elementary Matrices, aur Volume, Orientation & the Determinant.