3.5 Change of Basis · Linear Algebra Refresher

Once a basis is fixed, every vector becomes a list of numbers — its coordinate vector. Change the basis and the numbers change, even though the vector doesn't. A transition matrix is the dictionary between two such coordinate systems.

Definition · Coordinate vector

Let

E = {u_{1}, \dots, u_{n}}

be an ordered basis for

V

. Every

v \in V

has a unique expansion

v = c_{1} u_{1} + \dots + c_{n} u_{n}

; the vector

[v]_{E} = (c_{1}, \dots, c_{n})^{T}

is the coordinate vector of

v

relative to

E

!Exam trap

Bases are ordered here. Reordering the basis permutes the coordinates, which changes the transition matrix. Also, in

R^{n}

the coordinates relative to the standard basis are the entries themselves — that's the only case where the vector and its coordinate vector look identical.

In $R^{n}$ everything is matrix algebra. Put the basis vectors in as columns: $U = (u_{1}, \dots, u_{n})$ . Then $v = U c$ converts coordinates $\to$ vector, and $c = U^{- 1} v$ converts vector $\to$ coordinates. $U$ is nonsingular precisely because its columns are a basis.

coords in U [v]_{U} = S U^{- 1} V coords in V [v]_{V}

Definition · Transition matrix

S = U^{- 1} V

is the transition matrix from the ordered basis

V = {v_{1}, \dots, v_{n}}

U = {u_{1}, \dots, u_{n}}

. Its

j

th column is

[v_{j}]_{U}

— the old basis vectors written in the new coordinates.

→Shortcut

In practice, don't invert. Row reduce the augmented matrix

(U ∣ V)

(I ∣ S)

— this solves all

n

systems

U s_{j} = v_{j}

at once, exactly as in §1.5.

Worked example

Building a transition matrix

0/4 steps

Let

u_{1} = (1, 1)^{T}

u_{2} = (- 1, 1)^{T}

and

v_{1} = (2, 1)^{T}

v_{2} = (1, 0)^{T}

. Find the transition matrix from

{v_{1}, v_{2}}

{u_{1}, u_{2}}

∞Connection

For a general vector space, replace matrices with coordinate maps: to go from basis

E

to basis

F

, write each

e_{j}

in terms of the

f_{i}

and stack those coordinate vectors as columns. This is exactly the machinery §4.2 uses for matrix representations, and §6.1's 'similar matrices'

S^{- 1} A S

is the same idea applied to an operator.

Check yourself

U = (u_{1}, u_{2})

and

V = (v_{1}, v_{2})

hold two bases of

R^{2}

as columns. If

c

is the coordinate vector of

x

relative to

{v_{1}, v_{2}}

, its coordinate vector relative to

{u_{1}, u_{2}}

is:

T / F

Every transition matrix is nonsingular.

3.5Change of Basis

What you must be able to do

Building a transition matrix

Check yourself