4.2 Matrix Representations · Linear Algebra Refresher

Every linear transformation between finite-dimensional spaces is a matrix — you just have to say relative to which bases. Change the bases and the matrix changes; the transformation doesn't. That tension is what similarity and diagonalization exploit in Chapter 6.

Theorem 4.2.1 · Standard matrix

L : R^{n} \to R^{m}

is linear, there is a unique

m \times n

matrix

A

with

L (x) = A x

for all

x

, namely

A = (L (e_{1}), L (e_{2}), \dots, L (e_{n})) .

Why: $x = \sum x_{j} e_{j}$ , so $L (x) = \sum x_{j} L (e_{j})$ — a linear combination of the columns, which is exactly $A x$ .

Theorem 4.2.3 · General bases

Let

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

be an ordered basis for

V

and

F = {b_{1}, \dots, b_{m}}

for

W

. For a linear

L : V \to W

there is a unique

m \times n

matrix

A

with

[L (v)]_{F} = A [v]_{E}

, and the

j

th column of

A

[L (u_{j})]_{F}

★Key idea

Both bases matter, and they play different roles. $E$ decides what you feed in (apply

L

to each

u_{j}

); $F$ decides how you record the output (express each result in

F

-coordinates). Getting this backwards is the standard mistake.

Corollary 4.2.4 · The computational shortcut

For

L : R^{n} \to R^{m}

with bases

E

and

F = {b_{1}, \dots, b_{m}}

, the matrix representation

A

is obtained by row reducing

(b_{1}, \dots, b_{m} ∣ L (u_{1}), \dots, L (u_{n})) ⟶ (I ∣ A) .

It's the $(A ∣ I) \to (I ∣ A^{- 1})$ trick again — solving $n$ systems with the same coefficient matrix simultaneously.

Building a matrix representation

1
Identify the two bases
$E = {u_{j}}$ for the domain, $F = {b_{i}}$ for the codomain. If none is stated, both are standard and you're in the easy case.
2
Apply $L$ to each domain basis vector
Compute $L (u_{1}), \dots, L (u_{n})$ as ordinary vectors in $W$ .
3
Express each result in $F$ -coordinates
F standardNothing to do — the vectors are their coordinates. Stack them as columns.
F generalRow reduce $(b_{1}, \dots, b_{m} ∣ L (u_{1}), \dots, L (u_{n}))$ to $(I ∣ A)$ .
4
Sanity check on one vector
Pick a $v$ , compute $L (v)$ directly, and compare with $A [v]_{E}$ re-expanded in $F$ . Two minutes here catches sign and ordering errors.

Worked example

Same operator, non-standard bases

0/5 steps

Let

L (x) = (x_{1} + x_{2}, 2 x_{1} - x_{2})^{T}

R^{2}

. Find its matrix relative to

E = {u_{1} = (1, 2)^{T}, u_{2} = (3, 1)^{T}}

in the domain and

F = {b_{1} = (1, 0)^{T}, b_{2} = (1, 1)^{T}}

in the codomain.

∞Connection

V = W

and you change basis with transition matrix

S

, the operator's matrix changes from

A

S^{- 1} A S

— the definition of similar matrices. Theorem 6.1.1 then says similar matrices share a characteristic polynomial, which is exactly the statement that eigenvalues are a property of the operator, not of the coordinates.

Check yourself

To build the standard matrix of a linear transformation

L : R^{n} \to R^{m}

, the

j

th column of

A

is:

You need the matrix of

L : R^{n} \to R^{m}

relative to bases

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

and

F = {b_{1}, \dots, b_{m}}

. The efficient route is:

T / F

The matrix of the counterclockwise rotation of

R^{2}

θ

(cos θ sin θ - sin θ cos θ)

T / F

The same linear transformation can be represented by different matrices depending on the bases chosen.

4.2Matrix Representations

What you must be able to do

Building a matrix representation

Same operator, non-standard bases

Check yourself