Systems of linear equations — matrix form Ax = b: Foundations

4.5.8 · D1 · Maths › Linear Algebra (Full) › Systems of linear equations — matrix form Ax = b

Parent note ko trust karne se pehle, tumhe use padhna aana chahiye. Yeh page kuch bhi assume nahi karta — yahan tak ki yeh bhi nahi ki "vector" kya hota hai. Hum har ek symbol build karte hain, order mein, har ek apne pehle wale pe lean karta hua.

0. Ek number, aur ek number ka naam

Hum floor ke neeche se shuru karte hain. Ek scalar bas ek ordinary number hai: $5$ , $- 2$ , $\frac{1}{3}$ . Hum ise "scalar" isliye bolte hain kyunki iska ek hi kaam hai — scale karna — doosri cheezein stretch ya shrink karna. Bas yahi matlab hai; "scale" word yaad rakho.

1. Subscripts — ek street pe house numbers

Jab hamare paas ek hi tarah ke kaafi numbers hote hain, toh hum 20 naaye letters nahi banate. Hum ek hi letter reuse karte hain aur har copy ko ek address dete hain jo neeche likha hota hai. Woh address ek subscript hai.

$x_{1}, x_{2}, x_{3}$ — teen alag unknown numbers, sab " $x$ family" ke.
$a_{ij}$ — do address: $i$ row hai, $j$ column hai. Ise "a-eye-jay" padho. Toh $a_{23}$ row 2, column 3 mein rehta hai.

2. Vector — numbers ki ek list jisme direction ho

Kai scalars ko ek column mein stack karo aur tumhe ek vector milega.

x = [x_{1} x_{2}]

Ise picture karne ke do tarike hain, dono sach hain aur dono baad mein kaam aayenge:

Ek list — numbers ki ek "to-do" ("2 scoops aata, 1 scoop cheeni").
Ek arrow — origin $(0, 0)$ se shuru hoke, coordinate $(x_{1}, x_{2})$ ki taraf point karta hua. Iska length aur direction picture hai.

Figure — Systems of linear equations — matrix form Ax = b

2a. Vectors ko Scale aur Add karna (do moves)

Vectors ke saath sirf do hi operations kiye jaate hain, aur baad ki har idea inhi se bani hai.

Scale: ek vector ko scalar $c$ se multiply karo — arrow ko $c$ factor se stretch karo ( $c < 0$ ho toh flip karo).

2 [13] = [26]

Add: arrows ko tip-to-tail rakho; sum woh arrow hai jo bilkul shuru se bilkul end tak jaata hai.

[10] + [01] = [11]

3. Dot product — ek number jo "matched-up sum" measure karta hai

Same length ke do vectors lo. Unhe entry-by-entry multiply karo, phir saare products jod do. Woh single output number dot product hai.

[a_{1} a_{2}] \cdot [x_{1} x_{2}] = a_{1} x_{1} + a_{2} x_{2} .

Yeh exact recipe kyun? Ek system ki ek equation dekho:

a_{1} x_{1} + a_{2} x_{2} = b_{1} .

Left side already coefficient list $(a_{1}, a_{2})$ aur unknown list $(x_{1}, x_{2})$ ka dot product hai. Toh dot product koi extra invention nahi hai — yeh woh machine hai jo coefficient row plus unknowns ko equation ki left-hand side mein convert karti hai. Isliye hi parent note kehta hai " $A x$ ki $i$ -th entry, row $i$ aur $x$ ka dot product hai."

4. $\sum$ — ek compact "in sab ko add karo"

Symbol $== \sum ==$ (Greek capital sigma) ka matlab hai sum. Iske teen parts hain:

j = 1 \sum n a_{ij} x_{j} = a_{i 1} x_{1} + a_{i 2} x_{2} + \dots + a_{in} x_{n} .

neeche ( $j = 1$ ): counter yahan se shuru hota hai,
upar ( $n$ ): counter yahan ruk jaata hai,
body ( $a_{ij} x_{j}$ ): har $j$ plug karo aur results add karo.

5. Matrix — numbers ka poora grid

Sari equations ke coefficient rows ko ek rectangle mein stack karo: ek matrix.

A = [a_{11} a_{21} a_{12} a_{22}]

Iska size $m \times n$ hai: $m$ rows (ek per equation), $n$ columns (ek per unknown). Hum $A \in R^{m \times n}$ likhte hain.

Isse slice karne ke do tarike hain, dono zaroori hain:

Rows ke through — har row ek equation ke liye coefficient list hai (row view ko feed karta hai).
Columns ke through — har column $a_{(j)}$ ek vector hai (column view ko feed karta hai).

6. Matrix times vector — show ka star

Ab pieces assemble karo. $A x$ compute karne ke liye: $A$ ki har row lo, use $x$ ke saath dot karo (§3), aur results ko ek naaye vector mein stack karo.

[a_{11} a_{21} a_{12} a_{22}] [x_{1} x_{2}] = [a_{11} x_{1} + a_{12} x_{2} a_{21} x_{1} + a_{22} x_{2}]

Toh $i$ -th entry exactly $(A x)_{i} = j = 1 \sum n a_{ij} x_{j}$ hai — row $i$ aur $x$ ka dot product. Dekho kaise §3, §4, §5 sab yahan converge karte hain.

Doosri picture — column view. Wahi product columns ki ek linear combination mein regroup ho jaata hai:

A x = x_{1} [a_{11} a_{21}] + x_{2} [a_{12} a_{22}] = x_{1} a_{(1)} + x_{2} a_{(2)} .

Ise aise padho: "column 1 ke $x_{1}$ scoops, plus column 2 ke $x_{2}$ scoops." Yahi scaling (§2a) phir adding (§2a) hai — kuch naaya nahi.

7. Equals sign, theek se samjho

$A x = b$ ka matlab hai ki vector $A x$ aur vector $b$ same list hain — entry 1, entry 1 ke barabar hai, entry 2, entry 2 ke barabar hai, aur aage bhi. Ek vector equation secretly ek saath $m$ scalar equations hold karta hai. Yahi "bookkeeping" hai jise parent note celebrate karta hai.

8. Woh words jo answer mein milenge

Yeh outcomes ke naam hain; abhi inhe master karne ki zaroorat nahi — linked notes woh kaam karte hain — lekin tumhe yeh words pehchanne chahiye.

Definition Solving ki vocabulary

Span — har woh vector jo kisi given set ko scaling-and-adding se reach kiya ja sake. Reachable region. → Column space and null space
Column space $Col (A)$ — $A$ ke columns ka span; saare possible $A x$ outputs ka set. Solution exist karne ke liye $b$ yahan hona chahiye. → Column space and null space
Null space — saare inputs $x$ jinke liye $A x = 0$ ; yeh "free wiggle" directions hain. → Column space and null space
Rank — genuinely independent rows/columns ki count. Yeh existence aur uniqueness decide karta hai. → Rank of a matrix
Linear independence — koi bhi vector, set ke doosre vectors ki scale-and-add mix nahi hai. → Linear independence
Determinant $det A$ — ek number (sirf square matrices ke liye) jo nonzero tab hota hai jab $A$ invertible ho. → Determinant
Inverse $A^{- 1}$ — "undo" matrix jo $x = A^{- 1} b$ deta hai. → Inverse of a matrix

Prerequisite map

Har arrow ka matlab hai "right box samajhne se pehle left box chahiye." Dekho kaise sab kuch $A x$ mein, phir $A x = b$ mein funnel hota hai.

Equipment checklist

Recall Self-test: kya tum har cheez ek saansh mein bol sakte ho?

Scalar kya hota hai? ::: Ek single real number jiska kaam cheezein scale karna hai. $a_{ij}$ ke do subscripts ka kya matlab hai? ::: Row $i$ , column $j$ — ek address, number "ij" nahi. Vectors pe sirf do operations kya hain? ::: Scaling (scalar se stretch/flip) aur adding (tip-to-tail). Dot product kaise compute karte hain? ::: Matching positions mein entries multiply karo, phir unhe ek scalar mein sum karo. $\sum_{j = 1}^{n} a_{ij} x_{j}$ spell out karne pe kya milta hai? ::: $a_{i 1} x_{1} + a_{i 2} x_{2} + \dots + a_{in} x_{n}$ . $A x$ ko padhne ke do tarike kya hain? ::: Rows ko $x$ ke saath dot karo, YA $A$ ke columns ka weighted sum. $A x = b$ actually kya assert karta hai? ::: Output vector $A x$ , $b$ se entry-by-entry match karta hai — ek line mein $m$ equations. $x = A^{- 1} b$ ko " $A$ se divide karna" kyun nahi keh sakte? ::: Matrices mein division nahi hoti; $A^{- 1}$ ek alag undo-matrix hai jo tabhi exist karta hai jab $A$ square ho aur $det A \neq = 0$ ho.

Ready ho? Toh wapas jaao parent topic pe aur teenon views seedhi English ki tarah padhenge.