Basis — definition, uniqueness of representation: Question bank

4.5.17 · D5 · Maths › Linear Algebra (Full) › Basis — definition, uniqueness of representation

True or false — justify

TF1. Koi bhi set jo

R^{3}

ko span kare woh

R^{3}

ka basis hai.

False — spanning se har vector ki koi representation milti hai, lekin agar set bahut bada ho toh woh dependent hoga, isliye coordinates unique nahi honge; e.g. chaar vectors

R^{3}

ko span kar sakte hain par kabhi basis nahi ban sakte.

TF2.

V

mein koi bhi linearly independent set,

V

ka basis hai.

False — independence uniqueness deta hai par reach nahi;

{e_{1}, e_{2}}

R^{3}

mein independent hai phir bhi

(0, 0, 1)

nahi bana sakta, isliye spanning fail ho jaati hai.

TF3. Kisi bhi given space

V

ke har basis mein vectors ki sankhya same hoti hai.

True — woh common size invariant dimension

dim V

hai (dekho Dimension); koi bhi basis chhota ya bada nahi ho sakta.

TF4. Ek single nonzero vector us line (1-dimensional subspace) ka basis hai jo woh span karta hai.

True — ek nonzero vector automatically independent hota hai aur apni line ko span karta hai, isliye woh us 1-D subspace ka basis hai.

TF5. Empty set

{}

zero space

{0}

ka basis hai.

True — empty set vacuously independent hai aur

{0}

ko span karta hai (empty sum

0

hai), isliye

dim {0} = 0

TF6. Agar kisi set mein exactly

dim V

vectors hain, toh woh automatically basis hai.

False — sahi count zaroori hai par kaafi nahi;

{(1, 0), (2, 0)}

mein

R^{2}

mein do vectors hain par dependent hai aur sirf ek line ko span karta hai.

TF7. Ek vector ke coordinates har basis mein same hote hain.

False — coordinates ek chosen basis ke relative addresses hain; same arrow

(4, 2)

standard basis mein

(4, 2)

hai lekin

{(1, 1), (1, - 1)}

mein

(3, 1)

hai (dekho Coordinate Vectors and Change of Basis).

TF8. Basis vectors ko reorder karne se woh basis rehta hai ya nahi, yeh badal jaata hai.

False — spanning aur independence order ki parwah nahi karte; lekin reordering coordinate entries ko permute zaroor karta hai, isliye address list badal jaati hai, chahe set phir bhi basis rahe.

TF9. Do alag bases ek hi vector ko same coordinates de sakte hain.

True — standard basis aur koi bhi basis jo

w

ko same tuple pe bheje, us ek vector ke liye match kar sakte hain, lekin generally addresses alag hote hain; ek vector ke liye equality ka matlab bases equal nahi hota.

TF10. Agar har vector ki kam se kam ek representation hai, toh set basis hona chahiye.

False — "sabke liye kam se kam ek" exactly spanning hai; tumhe "zyada se zyada ek" bhi chahiye, jo independence hai, tab jaake basis keh sakte ho.

Spot the error

SE1. "

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

R^{2}

ko span karta hai aur mujhe koi bhi vector likhne deta hai, isliye yeh basis hai."

Error hai uniqueness skip karna:

(2, 3) = 2 e_{1} + 3 e_{2}

aur

= e_{1} + 2 e_{2} + (1, 1)

, do addresses, isliye yeh spanning ke baad bhi basis nahi hai.

SE2. "Kyunki

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

independent hai, yeh

R^{3}

ka basis hai."

Yeh xy-plane subspace ka basis hai,

R^{3}

ka nahi — spanning fail hoti hai kyunki koi vector plane se bahar

(0, 0, 1)

tak nahi pahunch sakta.

SE3. "Uniqueness proof mein hum do representations subtract karte hain aur seedha conclude karte hain

c_{i} = d_{i}

Ek step missing hai: subtract karne se

\sum (c_{i} - d_{i}) v_{i} = 0

milta hai, aur tab independence har coefficient ko zero force karti hai, jisse

c_{i} = d_{i}

milta hai.

SE4. "

{1, x, x^{2}, x^{2}}

P_{2}

ka basis hai kyunki ismein

1, x, x^{2}

hai."

Repeated

x^{2}

set ko dependent banata hai (

x^{2} - x^{2} = 0

nonzero coefficients ke saath), isliye yeh basis nahi ban sakta, chahe

P_{2}

ko span kare.

SE5. "Vector

(0, 0)

ki ek basis mein infinitely many representations hain, kyunki

0 = \sum 0 v_{i}

trivially milta hai."

Ek genuine basis mein

0

ki exactly ek representation hoti hai, woh all-zero wali — yeh uniqueness hi independence ki definition hai; infinitely many ka matlab hoga set dependent hai.

SE6. "

{(2, 0), (0, 3)}

basis nahi hai kyunki entries

1

nahi hain."

Entries

1

honi zaroori nahi; yeh dono independent hain aur

R^{2}

ko span karte hain, isliye yeh ek bilkul valid basis hai — sirf vector ke coordinates scale honge.

SE7. "Basis mein ek aur independent-lagte vector ko add karne se woh basis rehta hai, bas thoda bada ho jaata hai."

Tum koi bhi vector basis mein add karke independent nahi reh sakte — basis already span karta hai, isliye naya vector purane waalon ka combination hai, aur enlarged set dependent ho jaata hai.

Why questions

WHY1. Spanning akele coordinate system dene mein kyun fail ho jaata hai?

Kyunki yeh sirf yeh promise karta hai ki har vector ki koi recipe hai; independence ke bina ek vector ki kai recipes ho sakti hain, isliye koi ek well-defined address nahi hota jise coordinates keh sakein.

WHY2. Uniqueness proof mein "do equal sums" ko "kuch

= 0

" mein kyun convert karte hain?

Kyunki independence ek statement hai zero vector ke barabar combinations ke baare mein; equality ko

\sum (c_{i} - d_{i}) v_{i} = 0

likhna hi us definition ko invoke karne deta hai.

WHY3. Ek basis mein exactly

dim V

vectors kyun hone chahiye — na zyada, na kam?

dim V

se kam span nahi kar sakte (itni directions nahi ki har jagah pahunchen);

dim V

se zyada dependent hone chahiye (extra directions redundant hain), aur sirf exact count dono ho sakta hai.

WHY4. Standard basis mein coordinates vector ki entries ke equal kyun hote hain?

Har

e_{i}

exactly ek slot on karta hai, isliye slot

i

banane ke liye coefficient precisely wahi entry hoti hai — basis "invisible" hi isliye lagti hai kyunki yeh special alignment hai.

WHY5. Representation ki uniqueness woh payoff kyun hai jo matrices use karne deti hai?

Ek unique address har abstract vector ko ek definite coordinate column mein badal deta hai; ek linear map phir un columns par predictably act karta hai (dekho Matrix of a Linear Map).

WHY6. Abstract spaces jaise

P_{2}

mein basis kyun ho sakta hai, jabki koi arrows draw nahi kar sakte?

Basis ko sirf vector-space operations chahiye (add, scale), geometry nahi;

{1, x, x^{2}}

independent aur spanning hai, isliye

P_{2}

coordinate-wise bilkul

R^{3}

jaisa behave karta hai.

WHY7. Uniqueness specifically

w = 0

par apply karne se independence kyun recover hoti hai?

0

ki obvious representation all-zero coefficients wali hai; uniqueness kehti hai yeh ek hi hai, aur "

\sum c_{i} v_{i} = 0 \Rightarrow

sab

c_{i} = 0

" precisely independence ki definition hai.

Edge cases

EC1. Kya

{0}

(single zero vector) kabhi basis ka part hota hai?

Kabhi nahi — zero vector khud se dependent hai (

1 \cdot 0 = 0

nonzero coefficient ke saath), isliye ise contain karne wala koi bhi set independence fail karta hai.

EC2. Zero space

{0}

ka basis kya hai, aur uski dimension kya hai?

Empty set uska basis hai aur

dim {0} = 0

, kyunki koi directions specify nahi karni hain aur

0

already wahan ka akela vector hai.

EC3. Kya ek infinite set basis ho sakta hai?

Haan, infinite-dimensional spaces ke liye (e.g.

{1, x, x^{2}, \dots}

sabhi polynomials ke liye), lekin ek finite-dimensional

V

ka har basis finite hota hai jisme exactly

dim V

vectors hote hain.

EC4.

R^{n}

mein, kya

n

vectors ka koi bhi set basis hai jab tak koi zero vector na ho?

Nahi — unhe independent hona chahiye;

n

nonzero vectors phir bhi dependent ho sakte hain (e.g. do same line par hain), aur tab woh na span karte hain, na unique coordinates dete hain.

EC5. Agar do vectors parallel hain (ek doosre ka scalar multiple hai), kya dono ek basis mein ho sakte hain?

Nahi — parallel vectors dependent hote hain, isliye zyada se zyada ek aa sakta hai; doosra ek redundant direction add karta hai jo uniqueness tod deta hai.

EC6. Kya ek vector ka sign swap karna (

v_{i} \to - v_{i}

) basis rehne deta hai?

Haan —

- v_{i}

same space ko span karta hai aur

v_{i}

se independent hai; set basis rehta hai, haan us vector ka coordinate sign flip ho jaata hai.

Connections

Linear Independence — uniqueness half jo yahan har trap test karta hai.
Span and Spanning Sets — existence half.
Dimension — kyun vectors ki count matter karti hai.
Coordinate Vectors and Change of Basis — kyun addresses basis par depend karte hain.
Subspaces — $V$ ke andar chhote spaces ke bases.
Matrix of a Linear Map — unique coordinates ka payoff.