4.5.13 · D1Null space (kernel) and column space (image) — basis, dimens
Ek matrix ek machine hai jo input vectors khati hai aur output vectors ugalti hai , aur poora topic do sawaalon ke baare mein hai: woh kaun se outputs reach kar sakti hai? (column space) aur kaun se inputs woh zero kar deti hai? (null space).
Baaki sab — vectors, spans, dimensions, RREF — bas woh vocabulary hai jo un do sentences ko precisely kehne ke liye chahiye.
Parent note padh pane se pehle, tumhe us mein use kiya gaya har symbol achhe se samajhna hoga. Yeh page unhe ek-ek karke, ek-dusre par depend karne ke order mein, zero se build karta hai. Yahan kuch bhi yeh nahi maanta ki tumne pehle linear algebra dekhi hai.
Ek vector bas numbers ki ek ordered list hai, column ke roop mein likhi jaati hai:
x = x 1 x 2 x 3 . x 1 , x 2 , x 3 components hain (individual numbers). Subscript bas ek name-tag hai jo batata hai ki number kis slot mein baitha hai.
2 components wala vector flat plane mein ek arrow hota hai, origin (point ( 0 , 0 ) ( x 1 , x 2 ) hi arrow ki destination hai.
Humein vectors is liye chahiye kyunki matrix machine ek vector andar leta hai aur ek vector bahar deta hai. Vectors nahi, machine nahi.
Do cheezein jo tum vectors ke saath kar sakte ho — aur yahi do operations poora game hai:
Definition Do legal moves
Scaling (khichna): har component ko ek number α α x = ( α x 1 , α x 2 , … ) Adding : do arrows ko tip-to-tail line up karo; sum final tip tak ka arrow hota hai: ( x 1 + y 1 , x 2 + y 2 , … )
Sirf yahi do kyun? Kyunki "linear" ka matlab literally hai "sirf scaling aur adding se bana hua." Is topic ka har idea scaling aur adding ke yahi do moves hain, kisi na kisi roop mein.
R n R saare real numbers — number line par har point. Superscript n "n .
Toh R 2 R 3 R n n
Ek matrix numbers ka ek rectangular grid hota hai. Hum A ∈ R m × n A m rows (horizontal lines) aur n columns (vertical lines) hain.
A = [ 1 3 2 0 4 5 ] ( m = 2 rows , n = 3 columns ) .
Intuition Shape ko "output-by-input" ke roop mein padhoColumns (n m ek row of column-vectors hain jo side by side khadi hain :
A = [ a 1 a 2 ⋯ a n ] . woh tasveer hai jis par poora topic chalta hai — ise yaad rakho.
Humein kyun parwaah hai ki har column R m un columns ke combinations hote hain. Columns machine ka poora "ingredients ka stock" hain.
Definition Matrix times vector, seedha tarikaJab tum matrix A x A columns ka ek weighted mix hota hai, jahan weights x A x = x 1 a 1 + x 2 a 2 + ⋯ + x n a n .
Intuition Yeh kaisa dikhta haix x 1 x 2 Dials ghuma'o aur jodo = scaling aur adding = linear. Yahi poori machine hai.
Note karo ki A x ek output vector hai , aur woh hamesha columns ka koi combination hota hai. Yahi ek fact column space ka seed hai.
Definition Linear combinationa 1 , … , a n linear combination c 1 a 1 + ⋯ + c n a n c i A x
Kuch vectors ka span unse banaye ja sakne wale saare linear combinations hain — poora reachable set.
Ek nonzero arrow ka span = usse guzarne wali poori line .
Alag-alag directions mein do arrows ka span = woh poori plane jo woh sweep karte hain.
Ek hi line par do arrows ka span = abhi bhi sirf wahi line (doosre ne kuch naya nahi diya).
Intuition Span column space ke liye sahi word kyun haiColumn space { A x : x ∈ R n } saare possible outputs . Kyunki har output columns ka linear combination hai, "saare outputs" = "columns ke saare combinations" = columns ka span . Yahi literally column space hai.
Zero vector 0 0 x = 0
Definition Linearly independentVectors ka ek set linearly independent hai agar koi bhi vector doosron ka combination nahi hai — koi redundant nahi hai. Equivalently, unhe combine karke 0 ja sakta hai, toh set dependent hai (koi "dead weight" hai).
Ek space ki dimension kisi bhi basis mein vectors ki sankhya hai — usmein kitne independent directions hain. Ek line ki dimension 1 hai, ek plane ki 2, aur aage bhi aise.
Intuition Topic is par kyun jeeta-maarta haiParent ke do headline results — dim Col ( A ) = r dim Null ( A ) = n − r dimension counts hain. "Dimension = basis ka size" ke bina, woh boxed formulas sirf symbols hain. Yahi woh ruler hai jisse poora topic maapata hai.
Definition Row reduction & RREFRow reduction legal row moves ki ek fixed recipe hai (rows swap karo, ek row ko scale karo, ek row ka multiple doosri mein jodo) jo ek matrix ko simplify karta hai bina A x = 0 RREF (reduced row-echelon form) hai: har leading entry ek 1 Row reduction & RREF .
Definition Pivot, free column, rank
Ek pivot RREF mein ek leading 1 pivot column mein ek hota hai.
Ek free column mein koi pivot nahi hota — yeh ek aisi variable se correspond karta hai jo tum freely set kar sakte ho.
Rank r r = dim Col ( A ) Rank of a matrix .
Intuition RREF tool of choice kyun haiHumein ek mechanical tarika chahiye "kaun se columns redundant hain?" aur "kaun se inputs zero ho jaate hain?" ka jawab dene ke liye, bina ankhon se dekhke andaza lagaye. RREF dono ke jawaab ek saath deta hai: pivot columns independent directions mark karte hain (Col ke liye), aur free columns collapse karne ki degrees of freedom mark karte hain (null space ke liye). Yeh parent note mein dono bases ke peechhe ki ek hi engine hai.
Common mistake "Free columns zero rows hain."Kyun sahi lagta hai: zero rows obvious 'leftover' bits hain. Kyun galat hai: free-ness columns (inputs) ki property hai, zero rows rows (dependent equations) ke baare mein hain. Nullity free columns count karta hai = n − r
Vectors: lists and arrows
R to the n: input and output rooms
Matrix A as columns side by side
Product Ax as a mix of columns
Span equals reachable outputs
Null space and Column space
Upar ki saari cheezein solid honi chahiye parent topic samajhne se pehle. Topic ke baad aate hain Rank–Nullity Theorem , Injective and surjective linear maps , Solving Ax=b , aur Four fundamental subspaces .
Daayein side dhako aur khud ko test karo. Agar koi bhi jawab fuzzy lage, toh upar woh section dobara padho.
Vector kya hota hai, saral shabdon mein? Numbers ki ek ordered list, origin se us point tak ek arrow ki tarah tasveer mein.
R n n n n
A ∈ R m × n m n
Ek matrix ko columns side by side khade hue kyun padha ja sakta hai? Kyunki A x R m
A x A x x 1 a 1 + ⋯ + x n a n
Linear combination kya hota hai? Scaled vectors ko joda hua koi bhi sum c 1 a 1 + ⋯ + c n a n
Kuch vectors ka span kya hota hai? Unke SAARE linear combinations ka set — reachable line/plane/space.
Vectors linearly independent kab hote hain? Jab koi bhi doosron ka combination nahi hota; sirf all-zero weights 0
Basis kya hota hai? Ek minimal set jo space ko span bhi kare aur independent bhi ho.
Dimension kya hoti hai? Space ke kisi bhi basis mein vectors ki sankhya.
Pivot kya hota hai, aur rank kya hoti hai? RREF mein ek leading 1; rank r
RREF kya preserve karta hai aur kya change karta hai? A x = 0
Linear independence and basis — §5–§6 ke ideas poori tarah.Rank of a matrix — rank §7 ka pivot count hai.Row reduction & RREF — §7 ki engine.Rank–Nullity Theorem — woh counting identity jiske liye topic build karta hai.Injective and surjective linear maps — trivial null space ⇔ injective.Solving Ax=b — solvable tab hi agar b Four fundamental subspaces — Col aur Null chaaon subspaces mein kahan hain. 