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3.V.1. Changing Representations of Vectors

Definition 1.1: Change of Basis Matrix

The change of basis matrixfor bases B, D V is the representation of the identity map id : V → V w.r.t. those bases.

Lemma 1.2: Changing Basis

Proof:

Alternatively,

→

A matrix changes bases iff it is nonsingular.

Proof : Bases changing matrix must be invertible, hence nonsingular.

Proof : (See Hefferon, p.239.)

Nonsingular matrix is row equivalent to I.

Hence, it equals to the product of elementary matrices, which can be shown to represent change of bases.

Corollary 1.5:

A matrix is nonsingular it represents the identity map w.r.t. some pair of bases.

Exercises 3.V.1.

1. Find the change of basis matrix for B, DR2.

(a) B= E2 , D= e2 , e1

(b) B= E2 ,

(c)

D= E2

(d)

2. Let p be a polynomial in P3 with

where B= 1+x, 1x, x2+x3, x2x3 . Find a basis Dsuch that

Example 2.1: Rotation by π/6 in x-y planet: R2 → R2

Let

→

Consider t : V → V with matrix representation T w.r.t. some basis.

If basis B s.t. T = tB→B is diagonal,

Then t and T are said to be diagonalizable.

Definition 2.3: Matrix Equivalent

Same-sized matrices Hand Hare matrix equivalent

if nonsingular matrices Pand Qs.t.

H= P H Q or H = P 1H Q 1

Corollary 2.4:

Matrix equivalent matrices represent the same map, w.r.t. appropriate pair of bases.

Matrix equivalence classes.

Elementary row operations can be represented by left-multiplication (H= P H ).

Elementary column operations can be represented by right-multiplication ( H= H Q ).

Matrix equivalent operations cantain both (H= P H Q ).

∴ row equivalent matrix equivalent

Example 2.5:

and

are matrix equivalent but not row equivalent.

Theorem 2.6: Block Partial-Identity Form

Any mn matrix of rank k is matrix equivalent to the mn matrix that is all zeros except that the first k diagonal entries are ones.

Proof:

Gauss-Jordan reduction plus column reduction.

Corollary 2.8 left-multiplication (: Matrix Equivalent and Rank

Two same-sized matrices are matrix equivalent iff they have the same rank.

That is, the matrix equivalence classes are characterized by rank.

Proof.

Two same-sized matrices with the same rank are equivalent to the same block partial-identity matrix.

Example 2.9:

The 22 matrices have only three possible ranks: 0, 1, or 2.

Thus there are 3 matrix-equivalence classes.

If a linear map left-multiplication (f : V n → W m is rank k,

then some bases B → D s.t. f acts like a projection Rn → Rm.

Exercises 3.V.2. left-multiplication (

1. Show that, where A is a nonsingular square matrix, if P and Q are nonsingular square matrices such that PAQ = I then QP = A1.

2. Are matrix equivalence classes closed under scalar multiplication? Addition?

3.

(a) If two matrices are matrix-equivalent and invertible, must their inverses be matrix-equivalent?

(b) If two matrices have matrix-equivalent inverses, must the two be matrix- equivalent?

(c) If two matrices are square and matrix-equivalent, must their squares be

matrix-equivalent?

(d) If two matrices are square and have matrix-equivalent squares, must they be matrix-equivalent?

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