Basis — definition, uniqueness of representation

1. From scratch: what two properties do we need?

We want a set $B = \{v_1, \dots, v_n\}$ inside a vector space $V$ to act like coordinate axes.

We must be able to build any vector from $B$ → this is spanning.
We must have no redundancy (no axis expressible from the others) → this is linear independence.

2. The central theorem (DERIVE it)

Proof of " $\Rightarrow$ ": basis gives uniqueness.

Existence. Spanning literally says some $w=\sum c_i v_i$ exists. ✔

Uniqueness. Suppose $w$ had two representations: $w = \sum_i c_i v_i \quad\text{and}\quad w = \sum_i d_i v_i.$ Subtract them (legal — same vector minus itself): $\mathbf 0 = w - w = \sum_i (c_i - d_i)\, v_i.$ Why this step? Because turning "two equal sums" into "something $=\mathbf 0$ " lets us invoke independence, which is a statement about combinations equaling zero.

By linear independence, the only way $\sum_i (c_i-d_i)v_i=\mathbf 0$ is if every coefficient is zero: $c_i - d_i = 0 \;\Rightarrow\; c_i = d_i \quad\forall i.$ So the two representations were the same. Uniqueness proved. ∎

Proof of " $\Leftarrow$ ": uniqueness gives basis.

Existence-for-all = spanning. ✔
Uniqueness applied to $w=\mathbf 0$ : one rep is $\mathbf 0=\sum 0\cdot v_i$ . Uniqueness says it's the only one, so $\sum c_i v_i=\mathbf 0\Rightarrow c_i=0$ → that's exactly independence. ✔ ∎

Figure — Basis — definition, uniqueness of representation

3. Worked examples

Worked example A non-standard basis (coordinates ≠ components)

Let $B=\{(1,1),(1,-1)\}$ in $\mathbb R^2$ . Find coordinates of $w=(4,2)$ . Solve $c_1(1,1)+c_2(1,-1)=(4,2)$ : $c_1+c_2=4,\qquad c_1-c_2=2.$ Why this step? Spanning gives existence; we set up the linear system the coordinates must satisfy. Add: $2c_1=6\Rightarrow c_1=3$ ; then $c_2=1$ . Coordinates $[w]_B=(3,1)$ . Why unique? The system has a unique solution because $B$ is independent (the matrix $\begin{smallmatrix}1&1\\1&-1\end{smallmatrix}$ is invertible). Different basis → different address for the same arrow.

4. Common mistakes (Steel-manned)

5. The 80/20 core

6. Flashcards

What two conditions define a basis?

Linear independence AND spanning of

V

Which basis property guarantees existence of a representation?

Spanning.

Which basis property guarantees uniqueness of a representation?

Linear independence.

State the uniqueness theorem.

B

is a basis

\iff

every

w\in V

is uniquely

w=\sum c_i v_i

In the uniqueness proof, after assuming two reps, what trick is used?

Subtract them to get

\sum(c_i-d_i)v_i=\mathbf 0

, then apply independence.

What are the

c_i

w=\sum c_i v_i

called?

The coordinates of

w

relative to basis

B

Coordinates of

(4,2)

in basis

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

(3,1)

Why isn't

\{e_1,e_2,(1,1)\}

a basis of

\mathbb R^2

It spans but is dependent, so coordinates aren't unique.

Why isn't

\{e_1,e_2\}

a basis of

\mathbb R^3

Independent but doesn't span;

(0,0,1)

is unreachable.

How does uniqueness for

w=\mathbf 0

recover independence?

Only rep of

\mathbf 0

is all-zero coeffs, which IS the definition of independence.

What is the size of any basis equal to?

\dim V

Recall Feynman: explain to a 12-year-old

Imagine giving directions on a grid using only "steps East" and "steps North." Those two moves are your basis. Spanning means you can reach any spot. Independence means North isn't secretly the same as East — they're genuinely different directions. Because of that, every spot has one honest answer like "3 East, 1 North." If I gave you a third useless direction (like "Northeast"), suddenly the same spot could be described in many ways — confusing! A basis is the cleanest possible set of directions: enough to reach everywhere, but no leftovers, so every place has exactly one name.

Connections

Linear Independence — supplies the uniqueness half.
Span and Spanning Sets — supplies the existence half.
Dimension — the invariant size of every basis.
Coordinate Vectors and Change of Basis — coordinates $[w]_B$ depend on chosen $B$ .
Subspaces — bases let us measure subspace dimension.
Matrix of a Linear Map — built by feeding basis vectors through a map.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, basis ka matlab hai ek vector space ke liye sabse saaf-suthra coordinate system. Iske do hi conditions hai: pehla linear independence (koi bhi vector dusron se banaya na ja sake, yaani koi fizool direction nahi), aur dusra spanning (in vectors se space ka har element ban jaaye). Jab dono saath ho, tab kuch magical hota hai: har vector ka exactly ek hi address ban jaata hai — yaani unique coordinates.

Yeh uniqueness kahan se aati hai? Maano ek vector $w$ ke do alag-alag representations hai: $w=\sum c_i v_i$ aur $w=\sum d_i v_i$ . Inko subtract karo to $\mathbf 0=\sum(c_i-d_i)v_i$ milta hai. Ab independence kehta hai ki zero banane ka sirf ek tareeka hai — saare coefficients zero. Iska matlab $c_i=d_i$ , yaani dono representations same the. Bas yahi proof hai! Spanning se "kam se kam ek" representation milta hai (existence), independence se "zyada se zyada ek" (uniqueness) — milke "bilkul ek".

Common galti: students sochte hai ki sirf spanning kaafi hai, ya sirf independence kaafi hai. Agar bahut zyada vectors le lo (jaise $e_1,e_2,(1,1)$ in $\mathbb R^2$ ), to space to ban jaata hai par ek vector ke kai addresses ho jaate hai — yeh basis nahi. Aur agar kam vectors le lo (jaise sirf $e_1,e_2$ in $\mathbb R^3$ ), to kuch vectors reach hi nahi hote. Toh size bilkul $\dim V$ hona chahiye.

Yeh chapter isliye important hai kyunki ek baar basis fix kar lo, har abstract vector ek number-column ban jaata hai, aur phir poori linear algebra matrices se ho jaati hai — change of basis, transformations, sab kuch. Yaad rakho: S.I. se E.U. — Span + Independence dete hai Existence + Uniqueness.

Basis — definition, uniqueness of representation

1. From scratch: what two properties do we need?

2. The central theorem (DERIVE it)

3. Worked examples

4. Common mistakes (Steel-manned)

5. The 80/20 core

6. Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Linear Algebra (Full)

Connections