4.5.28 · D5 · HinglishLinear Algebra (Full)
Question bank — Matrix representation of linear transformations
4.5.28 · D5· Maths › Linear Algebra (Full) › Linear transformations ka matrix representation
Ye bank parent note Matrix representation of linear transformations par lean karta hai aur ideas ko poke karta hai Coordinate vectors and bases, Change of basis, Composition of linear maps, Rank and nullity, Kernel and image, aur Eigenvalues and diagonalization se.
True or false — justify karo
Matrix sirf linear map ki property hai.
False. Ye dono chosen bases aur par depend karta hai; same ke infinitely many matrices hote hain, ek har pair of bases ke liye.
Agar do matrices same linear map ko represent karte hain (alag bases ke through), to wo equal hone chahiye.
False. Wo equal nahi hone chahiye — wo equivalent hote hain (ya similar, agar saath badle), change-of-basis matrices ke through related hote hain. Dekho Change of basis.
Ek linear map jisme , ho, hamesha ek matrix rakhta hai.
True. Columns domain basis vectors count karte hain () aur rows codomain coordinates count karte hain (), isliye shape fix rehta hai chahe koi bhi bases chuno.
ke teen arbitrary vectors par ko jaanna hamesha ko completely determine karta hai.
False. Sirf tab jab wo teen vectors ek basis banate hain (linearly independent hain). Agar wo dependent hain, to un directions par undetermined hai jo unse miss hoti hain.
Zero map ka matrix har bases ki choice mein zero matrix hota hai.
True. Har basis vector par map hota hai, jiska coordinate vector kisi bhi codomain basis mein all zeros hota hai, isliye saare columns zero hain.
Sirf codomain basis badalna matrix ko change kar sakta hai jabki unchanged rahe.
True. Columns ko -basis mein store karte hain, isliye output ko recoordinatize karna har entry ko rewrite kar deta hai chahe inputs untouched hon.
Agar identity matrix hai, to identity map hona chahiye.
False. sirf yeh kehta hai ki har ka coordinate list jaisa hi hai; ke saath ye genuinely ek change-of-basis map ho sakta hai, identity nahi.
ka matrix hai (natural reading order mein).
False. Ye hai: pehle apply hota hai, isliye uska matrix right par hota hai, function composition order ko match karta hua. Dekho Composition of linear maps.
outputs ko rows mein likhne se ka valid matrix milta hai.
False. Isse transpose milta hai, jo ek alag map represent karta hai; formula outputs ko columns mein force karta hai.
Ek matrix jisme columns rows se zyada hain (), ek injective map represent kar sakta hai.
False. Injectivity ke liye zero kernel chahiye, lekin ek matrix jisme ho uski rank hoti hai, isliye uski nullity positive hai — koi nonzero input zero par map hota hai. Dekho Rank and nullity.
Error dhundo
" linear hai, isliye main iska matrix banaunga."
Error: linear nahi hai — . Koi linear-map matrix exist nahi karta; sirf linear maps ka matrix representation hota hai.
" hai isliye -wa column hai." (codomain basis )
Error: standard basis mein likha hai, mein nahi. Pehle solve karna padega, jo column deta hai.
", isliye iska matrix hai."
Error: rows = , columns = , isliye matrix hai, nahi.
"Kyunki rotate karta hai, iska matrix kisi bhi basis mein hai."
Error: wo matrix sirf standard basis mein hold karta hai. Ek skewed basis mein same rotation alag dikhta hai (parent ka Example 3 yahi dikhata hai).
" derivative ke liye, isliye main wo column chhod dunga."
Error: zero output bhi ek column hai — coordinate vector . Use drop karna shape toad deta hai aur matrix–vector product ko misalign kar deta hai.
" exist karta hai, isliye main aur kisi bhi order mein multiply kar sakta hun."
Error: intermediate space match hona chahiye — ke liye ka codomain basis , ke domain basis ke barabar hona chahiye. Doosra order conformable bhi nahi ho sakta.
"Map ka ek nonzero matrix hai, isliye wo invertible hai."
Error: nonzero invertible. Invertibility ke liye square, full-rank matrix chahiye; nonzero hai lekin square bhi nahi hai.
Why questions
ko codomain basis mein kyun express karna padta hai, domain basis mein nahi?
Kyunki mein rehta hai, aur coordinates sirf usi space ke basis ke relative sense banate hain jisme vector actually hota hai — isliye hum use karte hain, jo ka basis hai.
Matrix multiplication "row times column" rule kyun use karta hai, kuch simpler ke bajaye?
Ye reverse-engineered hai taaki do matrices ko multiply karna do maps ko compose karne ke barabar ho; rule wo price hai jo hold karne ke liye pay ki jaati hai. Dekho Composition of linear maps.
Ek linear map completely uski basis par values se kyun fix ho jaata hai?
Har vector ek unique combination hota hai, aur linearity deti hai — isliye finite data ko har jagah determine karta hai.
Same map ek basis mein diagonal aur doosri mein full kyun dikh sakta hai?
Ek diagonalizing basis map ke eigenvectors ke saath align karta hai, isliye har basis vector sirf scale hota hai — off-diagonal entries vanish ho jaati hain. Ek generic basis mein directions mix ho jaati hain. Dekho Eigenvalues and diagonalization.
Hum outputs columns mein kyun store karte hain, rows mein nahi?
Taaki (jo column select karta hai) exactly return kare, aur construction se sahi nikle.
ka kernel matrix ke null space ke same kyun hai?
Ek vector satisfies karta hai iff uska coordinate vector satisfies karta hai , isliye kernel aur null space coordinate map ke through one-to-one correspond karte hain. Dekho Kernel and image.
Matrix ka rank bases switch karne par kyun nahi badalta?
Change of basis invertible matrices se multiply karta hai, jo image ki dimension preserve karte hain; rank usi dimension ko measure karta hai, jo ki ek intrinsic property hai. Dekho Rank and nullity.
Edge cases
(zero space mein map) ka matrix kya hai?
Ye unique matrix hai — zero rows kyunki — jo us map ko represent karta hai jo sab kuch annihilate kar deta hai.
Kya ek -dimensional space se linear map ka matrix ho sakta hai?
Haan: ek matrix (koi columns nahi). -dimensional domain mein sirf ek vector hota hai, aur , jo empty product ke saath consistent hai.
Agar lekin invertible nahi hai, to kya matrix phir bhi square hai?
Haan, matrix square () hai kyunki ye basis vectors count karta hai, invertibility nahi. Ye sirf ek singular square matrix hai (determinant zero).
mein saare zeros wala column kya bata hai?
Corresponding basis vector par map hota hai, isliye ke kernel mein hai — Kernel and image ka direct read-off.
Agar ke basis vectors ka order badal do to matrix ka kya hoga?
Uske columns same order mein permute ho jaate hain; map unchanged rehta hai lekin uska bookkeeping table shuffle ho jaata hai — ek aur reminder ki matrix basis-dependent hai.
Agar codomain basis ka order badal do to matrix ka kya hoga?
Uske rows permute ho jaate hain, kyunki har column ki coordinate list ke naye order se match karne ke liye reindex ho jaati hai.
Identity map ke liye, same basis in aur out mein, matrix kya hai?
Identity matrix : har khud par map hota hai, jiske coordinates mein hain, diagonal ko ones se fill karte hain. Ek alag output basis choose karna instead ek change-of-basis matrix deta hai.
Recall Har trap ki one-line summary
Matrix ek basis-dependent bookkeeping table hai: columns codomain basis mein outputs pakdte hain, shape dimensions se aati hai, aur intrinsic facts (rank, kernel, invertibility) basis changes mein survive karte hain jabki entries khud nahi karti.
Connections
- Change of basis — entries kyun badte hain lekin map nahi.
- Composition of linear maps — multiplication order ki origin.
- Coordinate vectors and bases — wo jis par har trap rely karta hai.
- Rank and nullity — matrix se read-off hone wale basis-invariant facts.
- Kernel and image — zero columns aur null space.
- Eigenvalues and diagonalization — kyun kuch bases ko simplify karte hain.