4.5.27 · D1Linear transformations — definition, kernel, image
Ek linear transformation ek aisi machine hai jo poore arrows (vectors) ko idhar-udhar move karti hai, lekin do vaade nibhate hue: pehle add karke move karna, aur pehle move karke add karna — dono ek hi result dete hain; aur ek arrow ko stretch karna matlab uska output bhi utna hi stretch hoga. Parent page ki baaki sab cheezein — kernel, image, rank, nullity — bas yeh poochh rahi hain ki kaun se arrows zero mein crush ho jaate hain aur kaun se arrows machine produce kar sakti hai .
Parent note padhne se pehle, symbols ka ek poora toolbox quietly assume kiya jaata hai. Yeh page har ek cheez ko zero se build karti hai, ek aisi order mein jahan har idea sirf usse pehle wali cheezoon par tikaa ho.
Ek vector ek aisa arrow hai jiske paas ek length aur ek direction hoti hai, jo ek fixed point se shuru hota hai jise origin kehte hain. Coordinates mein hum ise numbers ki list ke roop mein likhte hain, jaise v = ( 3 , 2 ) 3 2
kyun arrows hain
Pair ( 3 , 2 ) 3 2 object ke do naam hain. Coordinates se hum calculate kar sakte hain; arrow se hum dekh sakte hain.
Figure s01 bilkul yahi dikhata hai: lavender arrow v = ( 3 , 2 ) 3 2
Yeh isliye zaroori hai kyunki ek linear transformation ek aisi machine hai jiske inputs aur outputs arrows hote hain . Arrows nahin, toh transform karne ko kuch nahin.
Arrows ko sirf do tareekon se combine kiya ja sakta hai, aur ek linear map in dono tareekon ko respect karne se hi define hoti hai.
Definition Vectors ko add karna (tip-to-tail)u + v u v u tip se start karo. Origin se final tip tak ka arrow u + v ( a , b ) + ( c , d ) = ( a + c , b + d )
Definition Vector ko scale karnac v c v c c = 2 c = 2 1 c = − 1 c ( a , b ) = ( c a , c b )
Figure s02 dono moves dikhata hai. Left side par, lavender u v u + v hai . Right side par, ek lavender arrow w 2 w − w
Intuition Exactly yahi do kyun?Arrows ki ek space in do moves se build hoti hai — har arrow jo tum reach kar sako woh doosron ki koi scaling-and-adding hai. Toh ek machine jo "structure respect kare" ko sirf yeh do operations respect karni hogi. Exactly yahi additivity (T ( u + v ) = T ( u ) + T ( v ) homogeneity (T ( c v ) = c T ( v )
Definition Scalar aur fieldF scalar (upar wala letter c Field F pool hai jin scalars se draw kiye jaate hain — numbers ka woh collection jisse tum scale kar sakte ho, aur jiske andar add, subtract, multiply aur divide (0 V , W F c ∈ F F c F = R c ∈ F c F
Definition Vector space aurR n vector space un saare arrows ka collection hai jo tum upar wali do operations se build kar sakte ho. Hum spaces ko capital letters V W R 2 2 R 3 3 R n n
∈ ⊆ { }
v ∈ V v set V " — arrow v V { x : condition } un saare x ka set jaise ki condition hold kare". Colon ka matlab "such that" hai.A ⊆ B A B A B
Topic ko yeh kyun chahiye: kernel likha jaata hai ker T = { v ∈ V : T ( v ) = 0 } ∈ ker T ⊆ V ⊆
Definition Function (map)Ek function ek aisa rule hai jo har input leke exactly ek output deta hai. Hum T : V → W T V W T ( v ) T v
Intuition Machine wali pictureT V W arrow →
Figure s03 wohi box hai: lavender arrow "input v V T T ( v ) W T : V → W
Topic ko yeh kyun chahiye: "linear transformation" ek special kind of function hai. Pehle jaanna zaroori hai function kya hota hai, phir hi un do fairness rules ko add kiya ja sakta hai jo ise linear banati hain.
0 Zero vector 0 v + 0 = v
Definition Basis aur basis vectorse i basis arrows ka woh smallest set hai jisse har doosra arrow scaling aur adding ke zariye build kiya ja sake. R 2 e 1 = ( 1 , 0 ) e 2 = ( 0 , 1 ) ( 3 , 2 ) = 3 e 1 + 2 e 2
Figure s04 arrow ( 3 , 2 ) e 1 e 2 3 e 1 2 e 2 ( 3 , 2 )
Intuition Basis poore topic ka secret kyun haiParent note ka punchline — "ek linear map completely determined hoti hai is baat se ki woh basis vectors ke saath kya karta hai" — tab hi samajh aata hai jab tum jaante ho ki e 1 , e 2 span karte hain. Kyunki koi bhi x = x 1 e 1 + ⋯ + x n e n T ( x ) = x 1 T ( e 1 ) + ⋯ + x n T ( e n ) har arrow par jaante ho. Change of basis dekho jab tum alag basis choose karo.
Kuch arrows ka span woh har arrow hai jise tum unhe scale aur add karke reach kar sako. span {( 1 , − 2 , 1 )} ( 1 , − 2 , 1 )
Ek space ki dimension uske liye basis mein arrows ki sankhya hai — independent directions ka count. Ek line ki dimension 1 2 3
Ek subspace ek chhota vector space hai jo ek bade ke andar baitha hota hai, origin se guzarta hai aur do operations ke under closed hota hai (members ko add ya scale karne par tum andar hi rehte ho). Origin se guzarne wali ek line ya plane R 3
Topic ko yeh kyun chahiye: kernel aur image hamesha subspaces hoti hain, aur rank/nullity bas unki dimensions hain. Rank–Nullity Theorem aur Column space and null space dekho jahan yeh cheezein main event ban jaati hain.
Ek matrix A m × n m n columns vectors hain; T ( e 1 ) , T ( e 2 ) , … T
A x follow hoti hai jo humne pehle se build ki hain. Input ko basis mein likho: x = x 1 e 1 + ⋯ + x n e n A T A T ( e 1 ) , … , T ( e n ) A x = T ( x ) = T ( ∑ i x i e i ) = additivity ∑ i T ( x i e i ) = homogeneity ∑ i x i T ( e i ) = ∑ i x i ( column i ) . x i A x
Topic ko yeh kyun chahiye: parent claim karta hai "image = A x A x Matrix multiplication mein rehti hain. Injective surjective bijective (the "one-to-one / onto" words) aur Eigenvalues and eigenvectors (special arrows jinhein ek map sirf stretch karta hai) ke saath milke, yeh foundations toolbox complete karte hain.
Ab jab arrows, maps, subspaces aur dimension build ho gaye hain, parent note ke chaar headline symbols plain reading ban jaate hain.
Definition Kernel operatorker T ker T ker of T ", short for kernel ) inputs ke ek set ka naam hai: V T ker T = { v ∈ V : T ( v ) = 0 W } . input space V
Definition Image notationim T im T image of T ", T ( V ) T V outputs ke ek set ka naam hai: har woh arrow jo machine actually produce kar sakti hai.
im T = T ( V ) = { T ( v ) : v ∈ V } ⊆ W . W
ker = k ill ho gaye inputs (V im = m ade outputs (W nullity killed count karta hai, rank made count karta hai.
Kernel ker T inputs killed
Right side hide karo aur khud test karo. Agar koi shaky lage, toh upar woh section dobara padho.
T : V → W T V W
Zor se padho: v ∈ V "Vector v V
Zor se padho: { v : T ( v ) = 0 } "Un saare v T ( v )
Zor se padho: ker T ⊆ V "Kernel input space V
Har vector space jo do operations allow karta hai woh hain ___ aur ___. vectors ko add karna (tip-to-tail) aur kisi number se scale karna.
Field F Un scalars ka pool jinse scale karna allowed hai; yahan F = R
Zero vector, ek picture ke roop mein, kya hai? Zero length ka ek arrow jo origin par baitha hai, kahin point nahin karta.
R 2 e 1 e 2 e 1 = ( 1 , 0 ) e 2 = ( 0 , 1 )
T ( e i ) T Koi bhi x = ∑ x i e i T ( x ) = ∑ x i T ( e i )
A x A x = ∑ x i e i A x = ∑ x i T ( e i ) = ∑ x i ( column i )
ker T Un inputs ka set jo zero mein crush ho jaate hain; yeh input space V
im T T ( V ) Un saare reachable outputs ka set; yeh output space W
Rank aur nullity kin do sets ki ___ hain? Dimensions — im T ker T
Ek space ki dimension kya hoti hai? Basis mein arrows ki sankhya — independent directions ka count.
Koi set subspace kab hota hai? Jab usme origin ho aur woh adding aur scaling ke under khud ke andar closed ho.
