4.5.28 · D1Linear transformations ka matrix representation
Ek linear transformation ek aisi machine hai jo arrows ko arrows mein badal deti hai — bina seedhi lines ko tedha kiye ya spacing ko bigaade. Jab aap input aur output ke liye kuch "ruler arrows" (ek basis) par agree kar lete ho, toh aapko bas yeh record karna hota hai ki har ruler arrow kahaan land karta hai — woh chhota sa numbers ka table hi matrix hai, aur woh aapko predict karne deta hai ki baaki har arrow kahaan jaata hai.
Is page par assume kiya gaya hai ki aapne parent note ki koi bhi notation pehle nahi dekhi. Hum har symbol ko scratch se build karte hain, ek aisi order mein jahan har ek sirf pehle waale par lean karta hai. Upar se neeche padho.
Ek vector ek aisa arrow hai jisme length aur direction hoti hai. Flat plane mein hum ise do numbers se describe karte hain: kitna right, aur kitna upar. Hum likhte hain v = ( x , y ) ( 0 , 0 ) ( x , y )
Neeche ki figure mein ek vector aur usse naam dene waale do numbers dikhaye gaye hain. Burnt-orange arrow dekho: woh 3 2 ( 3 , 2 )
Intuition Numbers kyun, sirf pictures nahi?Arrow ki picture achhi lagti hai, lekin aap picture ke saath compute nahi kar sakte. Do numbers attach karne se hum arrows ko add, scale kar sakte hain, aur aakhirkar arithmetic use karke arrows ko machines mein feed kar sakte hain. Poora topic arrows ki pictures ko numbers ke columns mein translate karna hai.
Har linear cheez exactly do operations se bani hai.
Definition Addition aur scalar multiplication
Add karo do vectors: unhe tip-to-tail rakho; result woh arrow hai jo bilkul shuru se bilkul end tak jaata hai. Numbers mein, matching slots add karo: ( a , b ) + ( c , d ) = ( a + c , b + d ) Scale karo ek vector ko ek number c scalar kehte hain, bas ek ordinary number): arrow ko stretch ya shrink karo. Numbers mein, c ⋅ ( a , b ) = ( c a , c b ) c
Definition Linear combinationVectors v 1 , v 2 , … linear combination har ek ko scale karke add karne ka koi bhi result hai: c 1 v 1 + c 2 v 2 + ⋯ c 1 c 2
Kyun important hai: parent note ka central claim — "ek basis T har vector kuch chosen arrows ka linear combination hota hai . Isliye yeh idea pehle aana chahiye.
Kisi space ka basis arrows ka ek chhota sa set hota hai jisse space ka har vector unka linear combination exactly ek tarike se banta ho. Plane ka sabse jaana-pehchana basis hai e 1 = ( 1 , 0 ) e 2 = ( 0 , 1 )
Figure dikhati hai ki arrow ( 3 , 2 ) banta hai: teen e 1 e 2
Intuition "Exactly ek tarike se" kyun magic word haiAgar koi vector basis se do alag tareekon se banta, toh uske "coordinates" ambiguous hote aur koi bhi bookkeeping table kaam nahi karti. Uniqueness hi poore matrix idea ko possible banati hai. Yahi uniqueness hai jis par parent note rely karta hai jab woh likhta hai "unique coordinates x i
Basis sirf e 1 , e 2 {( 1 , 1 ) , ( 1 , − 1 )} Coordinate vectors and bases dekho.
Definition Coordinate vector[ v ] B B = { b 1 , … , b n } v = x 1 b 1 + ⋯ + x n b n x 1 , … , x n v B [ v ] B = x 1 ⋮ x n . B B
Worked example Same arrow, do coordinate columnsArrow ( 3 , 2 ) [ v ] e = ( 2 3 ) B ′ = {( 1 , 1 ) , ( 1 , − 1 )} ( 3 , 2 ) = a ( 1 , 1 ) + b ( 1 , − 1 ) a = 2.5 b = 0.5 [ v ] B ′ = ( 0.5 2.5 ) Same arrow, alag numbers — kyunki ruler badal gaya.
Common mistake Yeh sochna ki vectorhi uske coordinates hain
Kyun sahi lagta hai: hum almost hamesha standard ruler use karte hain, jahan ( 3 , 2 ) Fix: coordinates sirf basis ke relative exist karte hain. Subscript B
Ek vector space kisi bhi cheez ka collection hai jise aap usual rules follow karke add aur scale kar sako. Plane mein arrows ek banate hain. Polynomials jaise 4 + 5 x + 6 x 2
Kisi space ki dimension uske kisi bhi basis mein arrows ki count hai — independent "directions" ki gintti. Plane ki dimension 2 2 P 2 { 1 , x , x 2 } 3
Hum V input space ke liye aur W output space ke liye. Letters n = dim V m = dim W m n
T : V → W T machine (ek function) hai jo space V W T : V → W T V W T ( v ) v
T linear hai agar woh Section 2 ke sirf do moves ka respect kare:
T ( u + v ) = T ( u ) + T ( v ) , T ( c v ) = c T ( v ) . Picture: seedhi, evenly-spaced lines ka grid machine chalane ke baad bhi seedha aur evenly-spaced rehta hai — koi bending nahi, koi curving nahi. Sirf stretching, rotating, shearing, ya flipping.
Figure ek linear machine (90°
Intuition Linearity hi poora game kyun haiKyunki T T T ( x 1 b 1 + ⋯ + x n b n ) = x 1 T ( b 1 ) + ⋯ + x n T ( b n ) . n T har cheez par T
Ek matrix numbers ka ek rectangular grid hai. Hum do bases use karte hain: B V C W [ T ] C B B C j [ T ( b j ) ] C j
Definition Matrix times columnA [ v ] B A A row lo, uski entries ko column ki entries se pair karo, pairs multiply karo aur add karo. Ise us row aur column ka dot product kehte hain. Result ek naya column hai — output coordinates.
In pieces ki deeper study hai Composition of linear maps , Rank and nullity , Kernel and image , Eigenvalues and diagonalization , aur Change of basis mein. Parent Matrix representation of linear transformations par tab wapas jaao jab neeche ke symbols automatic lagne lagein.
Add tip to tail and scale
Coordinates and bracket v in B
Vector space V and W dimension
Matrix table T in C from B
Output coords = matrix times input coords
Test karo khud ko. Har line: answer cover karo, yaad karo, phir reveal karo.
Ek plane vector ko kaun se do numbers naam dete hain, aur unka kya matlab hai? Horizontal step aur vertical step origin se arrow ki tip tak, likha jaata hai ( x , y )
Woh do operations kaun si hain jinse sab kuch linear bana hai? Vectors add karna (tip-to-tail) aur ek vector ko ek number se scale karna.
v 1 , v 2 Koi bhi c 1 v 1 + c 2 v 2
Arrows ke ek set ko basis kya banata hai? Har vector unka linear combination exactly ek tarike se banta hai.
"Exactly ek tarike se" kyun important hai? Yeh coordinates ko unambiguous banata hai, jo ek single matrix ko kaam karne deta hai.
[ v ] B Basis B v
Kya ek vector apne coordinates ke barabar hota hai? Nahi — coordinates is par depend karte hain ki aapne kaun si basis choose ki; subscript drop karo aur meaning ambiguous ho jaati hai.
Kisi space ki dimension kya hai? Uske kisi bhi basis mein arrows ki count.
T : V → W T V W
T T ( u + v ) = T ( u ) + T ( v ) T ( c v ) = c T ( v )
Basis par T T Har vector basis vectors ka unique combination hota hai, aur linearity T
[ T ] C B j T ( b j ) j C
T : V → W dim V = n dim W = m m n
Fundamental relationship batao. [ T ( v ) ] C = [ T ] C B [ v ] B
