Basis — definition, uniqueness of representation: Visual walkthrough

Step 1 — What is a vector, and what does "combine" mean?

WHAT. A vector is just an arrow with a tail at the origin and a tip somewhere in the plane. In pictures below the plane is the sunset-coloured grid; the origin is the dot at the centre labelled $O$ .

WHY. Before we can talk about "building" a vector from other vectors, we must agree on the two moves we are allowed:

scale an arrow (stretch or shrink it, flip it if the number is negative),
add two arrows (put the tail of the second on the tip of the first — the "tip-to-tail" rule).

PICTURE. Below, the violet arrow $v_{1}$ and the orange arrow $v_{2}$ are our two building blocks. The magenta dashed arrow shows $2 v_{1}$ (twice as long). The navy arrow shows $v_{1} + v_{2}$ built tip-to-tail.

Step 2 — "Spanning": can we reach everywhere?

WHAT. We ask: starting from our building blocks $v_{1}, v_{2}$ , which tips $w$ can we hit as we let $c_{1}, c_{2}$ range over all numbers?

WHY. A coordinate system is useless if some points have no address at all. Spanning is exactly the promise "every point has at least one address." We must check this first because you can't have a unique answer to a question that has no answer.

PICTURE. As $c_{1}$ and $c_{2}$ slide, the tip $w = c_{1} v_{1} + c_{2} v_{2}$ sweeps out the shaded region. When $v_{1}, v_{2}$ point in genuinely different directions, that region fills the entire plane — every target (the three magenta stars) is reachable.

Step 3 — The danger: a redundant building block

WHAT. Now add a third arrow $v_{3}$ that is not genuinely new — say $v_{3} = v_{1} + v_{2}$ . We still span the plane, but watch what happens to addresses.

WHY. We want to expose why spanning alone is not enough. The parent note's mistake box says this in words (" ${(1, 0), (0, 1), (1, 1)}$ spans but has two addresses"); here we see the two addresses for one point.

PICTURE. The single navy target point is reached two different ways:

green path: $2 v_{1} + 3 v_{2}$ (no $v_{3}$ ),
magenta path: $1 v_{1} + 2 v_{2} + 1 v_{3}$ . Same destination, two honest recipes. Redundancy has destroyed uniqueness.

Step 4 — "Independence": the one and only way to get zero

WHAT. We test our set against the single most important target: the zero vector $0$ (the arrow that goes nowhere, tail and tip both at $O$ ).

WHY. Here is the clever move. Instead of checking uniqueness at every point (infinitely many), we check it at one special point, $0$ . The set is linearly independent if the only recipe that lands you back at the origin is the do-nothing recipe $c_{1} = \dots = c_{n} = 0$ .

PICTURE. Left panel: for a redundant set, a non-zero recipe ( $1 v_{1} + 1 v_{2} - 1 v_{3}$ ) forms a closed loop back to $O$ — a "sneaky path to zero." Right panel: for an independent set, the only way the arrows cancel to $O$ is to not move at all.

Step 5 — Basis = both promises together

WHAT. A basis is a set that both spans (Step 2) and is independent (Step 4).

WHY. Combining the two promises:

Spanning $\Rightarrow$ at least one address (existence).
Independence $\Rightarrow$ at most one address (uniqueness — proven next).
Together $\Rightarrow$ exactly one address.

PICTURE. A clean sunset grid where $v_{1}, v_{2}$ are the two axes. Every point in the plane sits at the crossing of exactly one violet grid-line and one orange grid-line — its unique coordinate pair.

Step 6 — The uniqueness proof, drawn as a subtraction

WHAT. Suppose, for contradiction, that some vector $w$ had two addresses: $w = address A c_{1} v_{1} + \dots + c_{n} v_{n}, w = address B d_{1} v_{1} + \dots + d_{n} v_{n} .$

WHY subtract? We want to use independence, but independence is only a statement about combinations equal to $0$ . So we manufacture a " $= 0$ " statement by subtracting the two equal expressions: $0 = w - w = difference in slot 1 (c_{1} - d_{1}) v_{1} + \dots + difference in slot n (c_{n} - d_{n}) v_{n} .$

PICTURE. Geometrically: walking out along address A and then walking backwards along address B returns you to the origin — a closed loop. By Step 4, an independent set has no sneaky loop to zero, so every coefficient in the loop must be $0$ : $c_{i} - d_{i} = 0 ⟹ c_{i} = d_{i} for every i .$ The two addresses were identical all along. Uniqueness. ∎

Step 7 — Degenerate & edge cases (never leave the reader stranded)

WHAT. We check every scenario that "breaks" the picture, so nothing surprises you later.

WHY. A trustworthy coordinate system must be defined at all inputs, including the awkward ones.

PICTURE. Four panels:

Too few arrows — one arrow $v_{1}$ in the plane: it only reaches its own line. Points off the line (magenta star) have no address. Fails spanning. (parent: ${e_{1}, e_{2}}$ in $R^{3}$ .)
Too many arrows — three arrows in the plane: every point gets multiple addresses. Fails independence.
The zero vector as a building block — if some $v_{i} = 0$ , then $1 \cdot v_{i}$ is a sneaky non-trivial path to $0$ ; the set is automatically dependent, never a basis.
Coincident arrows — $v_{2}$ parallel to $v_{1}$ : they only span a line, not the plane. Fails spanning in 2D.

The one-picture summary

WHAT. Everything above, compressed: existence comes from spanning, uniqueness comes from independence, and their overlap is the single-address world of a basis.

Recall Feynman: the whole walkthrough in plain words

Imagine you are giving directions on a warm sunset grid using arrows. First (Step 1–2) you pick a couple of arrows and check you can reach every corner of town by stretching and stacking them — that's spanning, it means every place has at least one set of directions. But (Step 3) if you sneak in an extra arrow that's just "go the way of arrow 1 and arrow 2 combined," then the same house now has two sets of directions — confusing! To stop that, you demand (Step 4) that the only way your arrows cancel out back to the start is to not move at all — no sneaky loops home. That's independence, and it forces at most one set of directions per place. Put both promises together (Step 5) and every house has exactly one honest address. The proof that this really is unique (Step 6) is just: if a house had two addresses, walk out by the first and back by the second — you loop to the start, but independence forbids any loop-to-start except the empty one, so the two addresses were secretly the same. Finally (Step 7) we sanity-check the broken cases: too few arrows leaves parts of town unreachable, too many gives double-addresses, and a useless "stay-put" arrow is an instant sneaky loop. The sweet spot — exactly as many arrows as the town has dimensions — is a basis.

Connections

Linear Independence — Steps 3, 4, 6: the "no sneaky loop to zero" promise that forces uniqueness.
Span and Spanning Sets — Step 2: the "reach everywhere" promise that forces existence.
Dimension — Step 7: the exact number of arrows you must use.
Coordinate Vectors and Change of Basis — Step 5: the address $[w]_{B}$ each vector receives.
Subspaces — a line or plane through $O$ is spanned by its own smaller basis.
Matrix of a Linear Map — feeding basis arrows through a map builds its matrix columns.

Visual walkthrough — Basis — definition, uniqueness of representation