Foundations — Rank-nullity theorem — proof

4.5.14 · D1 · Maths › Linear Algebra (Full) › Rank-nullity theorem — proof

Theorem $\mathrm{rank}(T)+\mathrm{nullity}(T)=\dim V$ ko padhne se pehle bhi, tumhe pata hona chahiye ki un sabhi words aur symbols ka matlab kya hai. Yeh page unhe order mein banata hai — sabse saadhi idea se lekar poori statement tak. Yahan kuch bhi assume nahi kiya gaya hai — agar parent note ne kuch use kiya hai, toh hum use yahan define karte hain.

0. Vector kya hota hai? (the atom)

Hum yahan se kyun shuru karte hain? Kyunki sab kuch — spaces, maps, kernels — arrows se bana hai. Agar tum ek arrow visualize nahi kar sakte, toh baad ka koi bhi symbol real nahi lagega.

Woh do numbers vector ke components hain. Do vectors ko add karne ka matlab hai unhe tip-to-tail rakhna; ek vector ko number $c$ se scale karne ka matlab hai use stretch karna (ya flip karna, agar $c<0$ ho).

1. Vector space $V$ kya hai? (arrows se bhari ek room)

Theorem mein $V$ aur $W$ letters sirf do aisi rooms ke naam hain — input room aur output room.

Vectors sirf arrows nahi hone chahiye! Polynomials jaise $2x^2+x-5$ bhi arrows ki tarah add aur scale hote hain, isliye degree $\le 3$ ke saare polynomials ka set bhi ek vector space hai (parent ka Example 3). Isliye yeh theorem "abstract" spaces ke liye bhi kaam karta hai. Poori story ke liye Dimension of a Vector Space dekho.

2. Field $F$ (hum kis number se scale karte hain)

Tum mostly $F$ ko ignore karte ho — ise "ordinary numbers" samjho — lekin iska exist karna zaroori hai taaki "scale by $c$" ka matlab ho.

3. Linear independence & span (kya arrows overlap karte hain, ya space fill karte hain?)

Do ideas jinke upar poora proof tika hai.

4. Basis aur dimension (minimal complete kit)

Proof mein symbol $n$ sirf $\dim V$ ka short naam hai. "$V$ finite-dimensional" ka matlab hai ki $n$ ek finite number hai — tum ek basis list kar sakte ho aur ruk sakte ho.

"Ek subspace ke basis ko poori space ke basis tak extend karo" wala move Basis Extension Theorem hai, jo Step 1 ka engine hai.

5. Linear map $T:V\to W$ (the sorting machine)

Picture yeh hai: $T$ $V$ ki grid leti hai aur use $W$ mein stretch, rotate, ya flatten karti hai — lekin grid lines seedhi aur evenly spaced rakhti hai (linearity aisi dikhti hai). Kuch maps ek poori direction ko zero pe flatten kar deti hain — woh flattening kernel hai, next mein.

6. Zero vector $0$ aur kernel $\ker T$

Kernel input room $V$ ka ek subspace hai. Iske baare mein zyada Kernel and Image of a Linear Map mein hai.

7. Image $\mathrm{im}T$ aur rank

Image output room $W$ ka ek subspace hai. Ek crucial subtlety jo parent baar baar hammers karta hai: rank + nullity $\dim V$ (the domain) ke barabar hota hai, na ki $\dim W$ — kyunki hum count kar rahe hain ki input directions kaise sort hoti hain.

8. Matrices as linear maps

Ek matrix jaise $A=\left(\begin{array}{ccc}1& 2& 3\\ 2& 4& 6\end{array}\right)$ ek linear map hi hai: ise ek column vector $x$ dena aur $Ax$ compute karna ${\mathbb{R}}^3$ se ${\mathbb{R}}^2$ mein ek vector le jaata hai. $Ax=0$ solve karna kernel dhundhta hai; us solution mein free variables ki count nullity hai — woh link Solving Linear Systems — free variables count mein hai. Jab koi map koi bhi direction lose nahi karta (kernel sirf $\{0\}$ hai) toh woh ek isomorphism hai, Invertible Linear Maps and Isomorphisms dekho.

Pieces theorem ko kaise feed karte hain

Upar se neeche padho: arrows atoms hain, arrows se spaces aur independence bante hain, independence se bases bante hain, bases se dimension milta hai, aur kernel + image ka dimension exactly wahi hai jo theorem measure karta hai.

Equipment checklist

Khud test karo — right side cover karo aur zaur se jawab do.