Intuition The one core ideaA vector is a fixed arrow in space; a coordinate vector is just the shopping list of numbers you write once you pick a set of measuring-arrows (a basis). Change the measuring-arrows and the list changes, even though the arrow never moves — and the whole topic is the recipe for translating one list into another.
This page assumes you know nothing about the notation in the parent note. We build every symbol, one at a time, each on top of the last. Read top to bottom.
v vector is an arrow : it has a length and a direction. We write it with a bold letter, like v b 1 c 2
Look at the figure. The red arrow starts at the origin (the dot) and points somewhere. That is v we have not written any numbers yet. The arrow exists before we measure it — that is the single most important idea on this page.
Intuition Why "before numbers" mattersThe parent note keeps saying "the vector is invariant, only the numbers change." That sentence only makes sense once you accept the arrow is a real thing that lives in space independently of any numbers. The numbers are something we attach later.
A scalar is an ordinary number: 2 − 1 2 1 3 c 1 a α
Definition Column vector — the[ ] record an arrow as numbers, we stack the numbers vertically inside a tall bracket:
[ 4 2 ] . What the picture is: the top number is "how far right," the bottom number is "how far up." So [ 4 2 ] 4 steps right, then 2 steps up — and that lands you exactly at the tip of the arrow.
The tall bracket is just a container . It holds a list of scalars in a fixed order. The order matters: [ 4 2 ] [ 2 4 ]
Here is the subtle part the whole topic hinges on. "4 steps right, 2 steps up" secretly assumes what one step means . A step in which direction? By how much?
B = { b 1 , … , b n } basis is a chosen set of measuring-arrows. The curly script letter B C E names the set. The curly braces { … } b 1 , b 2 , …
span the space (its arrows can reach every point), andbe linearly independent (no arrow is redundant).
Look at the figure: the same target point can be reached with two different sets of measuring-arrows. Left uses the ordinary right/up arrows; right uses two slanted arrows. Same destination, different instructions.
Intuition Why we need a basis at allWithout agreed measuring-arrows, "the numbers of v Different basis → different list → this is the entire subject of change of basis. See Basis and dimension for why every basis of a space has the same count n
Subscript b j j name tag , not a power. b 1 , b 2 , b 3 n n = 2 n = 3 dimension (Basis and dimension ).The dots … : shorthand for "keep going in the obvious pattern." { b 1 , … , b n } b 1 b 2 b n
Definition Linear combinationTo build an arrow from the basis, you scale each measuring-arrow by a scalar and add the results:
v = c 1 b 1 + c 2 b 2 + ⋯ + c n b n . linear combination : "linear" because we only stretch (scale) and add — no bending, no squaring.
Definition The summation symbol∑ i = 1 ∑ n c i b i c i b i i 1 n Σ S um. It is just abbreviation — nothing new.
The picture shows building v c 1 b 1 c 2 b 2
Definition Coordinate vector[ v ] B c 1 , … , c n [ v ] B = c 1 ⋮ c n . Read the notation: the square brackets around v B v B ⋮
Intuition Why the list is unique — and why we careSuppose two lists ( c 1 , … ) ( c 1 ′ , … ) same v b i 0 independence says the only way to combine the b i 0 v
B R n n columns of a grid of numbers:
B = [ b 1 b 2 ⋯ b n ] . matrix is just a rectangle of scalars. Here each column is one measuring-arrow written out.
B does
Multiplying B [ v ] B j c j B [ v ] B = v B B
If B B backward .
Definition Inverse matrixB − 1 B − 1 B undoes B B B − 1 B − 1 B = I I identity matrix (the "do-nothing" machine: 1 0 B − 1 B Invertible matrices .
Common mistake "Every square matrix has an inverse."Why it feels right: small 2 × 2 The fix: if two columns of B B − 1 not exist. That is exactly why the topic insists B
With these two machines, the parent note's punchline reads cleanly:
C ← B P = C − 1 B — apply B (to standard), then C − 1 (standard to C ).
The little arrow C ← B P right-to-left : "input is B C Similar matrices and diagonalization you'll see the same C − 1 ( ⋅ ) B
Basis = chosen measuring arrows
Linear combination = scale and add
Coordinate list is unique
Matrix B = basis as columns
Change of basis P equals C inv B
Cover the right side and test yourself.
A bold letter v an arrow (a vector), which exists before any numbers are chosen.
A plain letter like c 1 α a scalar — an ordinary number that scales an arrow.
The tall bracket [ 4 2 ] a column of numbers, read top = right-steps, bottom = up-steps, in a fixed order.
A basis B = { b 1 , … , b n } a chosen set of measuring-arrows that span the space and are linearly independent.
The subscript j b j a name tag telling the arrows apart, NOT a power.
i = 1 ∑ n c i b i add up c i b i i = 1 n
The coordinate vector [ v ] B the unique column of scalars c i v = ∑ c i b i
Why is that column unique? because the basis is linearly independent, so only one combination builds v
The matrix B = [ b 1 ⋯ b n ] multiplies a B
B − 1 it undoes B B
Read C ← B P takes B C C − 1 B
Parent topic: Coordinate vectors — change of basis .
