Change of basis matrix: Foundations — Maths deep dive

Before you can build that machine, you must be fluent in the raw ingredients. Below is every symbol and idea the parent note leans on, in build-order. Nothing is used before it is drawn.

1. What is a vector? (the arrow that never moves)

The letter $v$ (or $b_{1}$ , $w$ , ...) is just a name for one such arrow.

Why the topic needs it: the whole point of "change of basis" is that this arrow stays put while our numbers for it change. If you forget the arrow is fixed, every sign and inverse goes wrong. See it in the figure: the amber arrow is the same in both panels.

2. The plane $R^{n}$ and the origin $O$

$O$ = the origin, the shared starting point of every arrow (where all axes cross).
$n$ = the dimension, the number of directions you need. In $R^{2}$ , $n = 2$ .

Why the topic needs it: the number of coordinates equals $n$ , and the change of basis matrix is an $n \times n$ square. See Basis and Dimension for where $n$ comes from.

3. Scaling and adding arrows

Two operations turn arrows into a language.

Why the topic needs it: every coordinate expansion is "scale some reference arrows, then add them." The formula $v = c_{1} b_{1} + c_{2} b_{2}$ is nothing but Step 1 (scale) followed by Step 2 (add), done repeatedly.

4. Linear combination

Reading the sigma shorthand. The big Greek "S" ( $Σ$ , sigma, for sum) packs that whole line: $\sum_{j = 1}^{n} c_{j} b_{j} = c_{1} b_{1} + c_{2} b_{2} + \dots + c_{n} b_{n} .$ The letter $j$ under $\sum$ is a counter that walks $1, 2, \dots, n$ ; for each value it grabs the $j$ -th scalar $c_{j}$ and the $j$ -th arrow $b_{j}$ , and you add all those pieces. It is only a compact way to write the long sum — nothing new.

Why the topic needs it: the parent's Steps 1–3 are drowning in $\sum$ signs. If $\sum$ is a mystery, the derivation is unreadable.

5. Linear independence & span → basis

Why the topic needs it: a basis is the only thing that guarantees every vector has exactly one coordinate description — not zero, not many. Independence kills duplicates; spanning guarantees coverage. Full detail lives in Basis and Dimension.

6. Coordinates and the column $[v]_{B}$

Decode the notation piece by piece:

$[v]_{B}$ — "the coordinate column of the arrow $v$ , measured with basis $B$ ." The little subscript $B$ names which ruler set you used.
The tall bracket $c_{1} ⋮ c_{n}$ — a column vector: numbers written top-to-bottom. The three dots $⋮$ mean "keep going the same way."

$e_{j}$ (used in the parent) is a special coordinate column: all zeros except a single $1$ in slot $j$ . Picture it as "one full step along direction $j$ , none along the others."

7. Matrix × column: the translator's engine

Here $p_{ij}$ means "the number in row $i$ , column $j$ " — row first, column second, always.

Why the topic needs it: the change of basis matrix acts by matrix–vector multiplication. The final Step 4 of the derivation is literally "recognize the sum as $P x$ ."

8. The inverse matrix $P^{- 1}$

For a $2 \times 2$ block there is a hand formula: $[a c b d]^{- 1} = \frac{1}{a d - b c} [d - c - b a],$ valid whenever the number $a d - b c$ (the determinant) is nonzero.

Why the topic needs it: "translate then translate back" is the identity, so the reverse dictionary is $P_{B \leftarrow C} = (P_{C \leftarrow B})^{- 1}$ . A basis is always independent, so its matrix always has nonzero determinant and the inverse always exists — that guarantee is Invertible Matrices.

fixed arrow (length + direction) from the origin, independent of any numbers.

R^{n}

means

all ordered lists of

n

real numbers — an

n

-dimensional space;

n

counts independent directions.

A scalar is

an ordinary real number that stretches (and possibly flips) an arrow.

To add two arrows you

place them tip-to-tail; the sum runs from the first tail to the last tip.

A linear combination is

c_{1} b_{1} + \dots + c_{n} b_{n}

— scale each arrow, then add.

\sum_{j = 1}^{n} c_{j} b_{j}

is shorthand for

the full sum

c_{1} b_{1} + c_{2} b_{2} + \dots + c_{n} b_{n}

;

j

is a counter.

A basis is

a set of arrows that is both linearly independent and spanning (minimal + complete).

Why a basis gives unique coordinates

independence forbids two different expansions of the same vector.

[v]_{B}

means

the column of scalars describing arrow

v

using basis

B

e_{j}

the coordinate column with a

1

in slot

j

and

0

elsewhere.

P x

computed by columns is

a linear combination of

P

's columns weighted by the entries of

x

;

P e_{j}

= column

j

P^{- 1}

satisfies

P^{- 1} P = P P^{- 1} = I

; it undoes

P

The

2 \times 2

inverse formula

\frac{1}{a d - b c} [d - c - b a]

, valid when

a d - b c \neq = 0

Passive vs active

passive = numbers change, arrow fixed (this topic); active = arrow actually moves.

Connections

Basis and Dimension — where the number $n$ and "independent + spanning" come from.
Coordinate Vectors — the columns $[v]_{B}$ we manipulate.
Invertible Matrices — guarantees $P^{- 1}$ always exists for a basis.
Linear Transformations — the active cousin of this passive story.
Similar Matrices — reuses $P^{- 1} (\cdot) P$ on actual maps.
Eigenvectors and Diagonalization — changing to a special basis.
Change of basis matrix — the parent this page equips you for.

Foundations — Change of basis matrix