Span — definition: Question bank — Hinglish deep dive

4.5.16 · D5 · Maths › Linear Algebra (Full) › Span — definition

True ya false — justify karo

True ya false: Ek span aisi line ho sakti hai jo origin se na guzare.

False. Sabhi scalars zero choose karne par hamesha

0

milta hai, isliye har span mein origin hota hai. Picture karo: ek off-origin line (jaise

y = x + 1

) points ka ek valid set hai, lekin tum

0

par "ghar" nahi reh sakte, isliye ye pehli hi membership check fail kar deti hai — niche figure mein red dashed line dekho, jo span nahi hai.

True ya false: Agar

v_{2} = 2 v_{1}

ho, toh

span {v_{1}, v_{2}}

ek plane hai.

False.

v_{2}

bilkul

v_{1}

ki hi line par hai (sirf do guna lamba), isliye unhe tip-to-tail rakhne par tum us line se bahar nahi jaate; span ek-dimensional rehti hai. Niche wala collapse figure dekho jahan dono arrows ek hi ray par baithe hain.

True ya false:

span {v_{1}, v_{2}} = span {v_{2}, v_{1}}

True. Arrows ka order badalne se jo tum build kar sakte ho uspe koi fark nahi padta — wahi tip-to-tail combinations dono taraf se exist karti hain, isliye reachable territory bilkul same rehti hai.

True ya false: Empty set ka span empty hota hai.

False.

{}

ka span

{0}

hai (empty combination zero deti hai), jo ki sabse chhota possible subspace hai — origin par ek single point, "kuch nahi" nahi.

True ya false: Agar

w \in span {v_{1}, v_{2}}

ho, toh

span {v_{1}, v_{2}, w} = span {v_{1}, v_{2}}

True.

w

pehle se hi current reachable region (plane ya line) ke andar baitha hai, isliye usse generator list mein daalne se koi nayi direction nahi milti; shape unchanged rehti hai.

True ya false:

R^{3}

mein teen vectors hamesha poore

R^{3}

ko span karte hain.

False. Ye

R^{3}

ko tabhi fill karte hain jab ye linearly independent hon. Agar teeno ek flat plane mein hon (coplanar), toh har combination usi plane mein rehti hai — tumhe ek 2D sheet milti hai, poori space nahi.

True ya false: Scalars ko

c_{i} \geq 0

tak restrict karne par bhi span wahi rehta hai.

False. Sirf nonnegative scalars ke saath tum kabhi "ulti taraf" nahi ja sakte, isliye origin se guzarne wali poori line ki jagah tumhe ek ek-taraf wala cone/wedge milta hai. Niche cone-vs-line panel dekho: span (green) dono taraf khulta hai; cone (orange) ek taraf.

True ya false: Ek span subtraction ke under closed hoti hai.

True. Subtract karna essentially ulta vector

(- 1) \cdot y

add karna hai; spans flipping aur adding dono ke under closed hoti hain, isliye tip-to-tail result

x - y

andar hi land karta hai.

True ya false:

R^{2}

mein nonzero

v

ke liye

span {v}

one-dimensional hai.

True. Ek akela nonzero arrow, stretch aur flip hone par, exactly origin se guzarne wali ek line sweep karta hai — ek free scalar, ek dimension.

True ya false: Agar

v_{1}, v_{2}

R^{2}

ko span karte hain, toh

v_{1}, v_{2}

columns wale matrix ka determinant zero hai.

False.

R^{2}

ko span karne ka matlab hai ki dono arrows alag directions mein point karte hain, isliye unse bana parallelogram nonzero area ka hai — aur area exactly

∣ det ∣

hai, jo isliye nonzero hai (dekho Determinant).

Error dhundho

Claim: "

span {v_{1}, \dots, v_{k}}

sirf set

{v_{1}, \dots, v_{k}}

hai."

Galat. Mental slip ye hai ki generators ko hi poora answer maanna. Lekin span woh infinite fill hai jo unhe stretch aur add karne se milti hai — ek line ya plane,

k

isolated dots nahi.

k

vectors seeds hain; span woh poora field hai jisme wo ugote hain (saari linear combinations).

Claim: "

(2, 2)

span {(2, 1)}

mein hai kyunki dono entries positive hain."

Galat. Slip ye hai ki "similar lagta hai / positive hai" ko "line par hai" samajhna.

(2, 1)

ka span sirf slope

\frac{1}{2}

wali ek line hai; point

(2, 2)

ka slope

1

hai, isliye ye us line se bahar hai. Algebra confirm karta hai:

c (2, 1) = (2, 2)

ko pehli entry se

c = 1

chahiye lekin phir doosri entry mein

1 \neq = 2

milta hai. Entries ka sign irrelevant hai — geometry yeh hai ki kaun si line hai.

Claim: "Koi bhi extra vector add karne se span hamesha bada ho jaata hai."

Galat. Slip: "zyada arrows = zyada reach." Lekin agar naya arrow pehle se current line/plane ke andar lie karta hai, toh wo koi nayi jagah point nahi karta, isliye shape bilkul nahi badlti. Collapse figure mein,

(2, 2)

{(1, 1)}

mein add karna ek line hi rakhta hai. Growth ke liye genuinely independent direction chahiye.

Claim: "

span {(1, 0, 0), (0, 1, 0)}

mein

(0, 0, 5)

hai kyunki hum freely scale kar sakte hain."

Galat. Ye dono arrows floor (

x y

-plane) mein flat hain, isliye har combination floor par hi rehti hai jisme third coordinate

a \cdot 0 + b \cdot 0 = 0

hai. Point

(0, 0, 5)

floor se seedha upar uda hua hai — unreachable. Free scaling tab bhi tumhe generators ki plane se bahar nahi nikaal sakta.

Claim: "

R^{3}

mein do vectors hamesha ek plane span karte hain."

Galat. Slip kitne vectors hain ginne ki jagah kitni directions hain count karna miss karna hai. Do arrows plane tabhi span karte hain jab wo alag directions mein point karein; agar ek doosre ka multiple ho (ya koi

0

ho), toh wo ek line share karte hain aur zyada se zyada usi line ko span karte hain — jaise collapse figure dikhata hai.

Claim: "

0

sirf tabhi span mein hota hai jab koi ek generator

0

ho."

Galat. Origin guaranteed hai chahe generators kuch bhi hon: saare scalars zero set karo aur tum

0

par land karte ho. Overview figure mein har span shape bilkul isi wajah se black origin dot se guzarti hai.

Why questions

Ek span hamesha origin se kyun guzarti hai?

Kyunki

c_{1} = \dots = c_{k} = 0

hamesha scalars ka ek valid choice hai, aur ye

0

deta hai — tum hamesha "na hilne" ka option choose kar sakte ho, aur exactly origin par land karte ho.

Span hamesha ek subspace kyun hoti hai, kabhi random blob nahi?

Isme

0

hota hai aur ye addition (

\sum a_{i} v_{i} + \sum b_{i} v_{i} = \sum (a_{i} + b_{i}) v_{i}

) aur scaling (

λ \sum a_{i} v_{i} = \sum (λ a_{i}) v_{i}

) dono ke under closed hai — exactly teen subspace conditions — isliye ye origin se guzarne wali ek perfectly flat sheet/line hai, kabhi bent ya floating blob nahi.

Hum sirf positive scalars ki jagah negative aur zero scalars kyun allow karte hain?

Negatives opposite direction tak pahunchne dete hain aur zeros kisi vector ko drop karne dete hain; milkar ye ek one-sided cone ko origin se guzarne wali poori symmetric flat mein badal dete hain — ye exactly woh fark hai jo cone-vs-line figure mein draw kiya gaya hai.

w \in span {v_{i}}

" test karna ek linear system solve karne tak kyun reduce ho jaata hai?

Sawaal "kya scalars

c_{i}

exist karte hain jisme

\sum c_{i} v_{i} = w

?" literally unknowns

c_{i}

mein linear equations ka ek set hai; solution exist karna matlab haan, koi nahi matlab nahi.

2 \times 2

matrix

[v_{1} v_{2}]

ka nonzero determinant ye kyun prove karta hai ki wo

R^{2}

span karte hain?

Nonzero determinant matlab matrix invertible hai, isliye

\sum c_{i} v_{i} = (x, y)

ka har target

(x, y)

ke liye ek (unique) solution hai — har vector reachable hai. Geometrically dono arrows ek real, non-flattened parallelogram enclose karte hain jiska area nonzero hai.

v_{2} = 2 v_{1}

span ko line tak kyun collapse kar deta hai?

Har combination

c_{1} v_{1} + c_{2} (2 v_{1}) = (c_{1} + 2 c_{2}) v_{1}

ban jaati hai, ek single scalar times

v_{1}

, isliye saare outputs

v_{1}

se guzarne wali ek line par hote hain.

Generators ki sankhya span ki dimension se same kyun nahi hoti?

Generators redundant (dependent) ho sakte hain; dimension sirf genuinely independent directions count karti hai, jo ek basis ka size hai — aksar generators ki sankhya se kam.

Edge cases

span {0}

(akela zero vector) kya hai?

Sirf

{0}

—

0

ka har scalar multiple phir bhi

0

hai, isliye span ek single-point origin subspace hai.

Vectors ke empty set ka span kya hai?

Zero subspace

{0}

, kyunki scaled vectors ka "empty sum"

0

define kiya jaata hai.

span {v}

kaisa dikhta hai agar

v \neq = 0

ho?

v

ki direction mein origin se guzarne wali ek line, jo infinitely dono taraf extend karti hai kyunki scalars positive ya negative dono ho sakte hain.

span {v_{1}, \dots, v_{k}}

kya hai agar har

v_{i} = 0

ho?

Phir bhi

{0}

; chahe kitne bhi zero vectors list karo, har combination zero hi hai.

Agar

v_{1}, v_{2}, v_{3} \in R^{2}

hon, toh kya unka span poora

R^{2}

ho sakta hai?

Haan —

R^{2}

mein teen vectors necessarily dependent hote hain, lekin unme se do pehle se independent ho sakte hain aur poori plane span kar sakte hain; extra vectors usse shrink nahi karte.

Kya

R^{3}

mein ek span exactly ek single point ho sakta hai?

Haan — sirf zero vectors (ya empty set) ka span

{0}

hai, origin par ek zero-dimensional point.

Agar ek generator baaki sab ki linear combination ho, toh usse delete karne par span badlta hai kya?

Nahi. Woh generator redundant tha, isliye usse hatane par same reachable set rehta hai — exactly isi tarah tum basis ki taraf prune karte ho.

Trap-spotter ki strategy, ek flowchart ke roop mein

Har span question ko in teen gates se order mein guzaro — diagram niche wale mnemonic ko mirror karta hai.

Connections

Linear combination — har membership test inhi se banta hai.
Linear independence — decide karta hai ki extra generators span badhaate hain ya nahi.
Subspace — har span basically yahi hoti hai.
Basis — span ka minimal, non-redundant generating set.
Dimension — raw generators nahi, independent directions count karta hai.
Determinant — square cases mein full-span test.
Column space — kisi matrix ke columns ka span.