1. 3.IV. Change of Basis 3.IV.1. Changing Representations of Vectors
3.IV.2. Changing Map Representations

2. 3.IV.1. Changing Representations of Vectors
Definition 1.1: Change of Basis Matrix
The change of basis matrix for 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,

3. Example 1.3: →

4. Lemma 1.4:
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.

5. Exercises 3.IV.1. 1. Find the change of basis matrix for B, D  R2.
(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, 1x, x2+x3, x2x3  . Find a basis D such that

6. 3.IV.2. Changing Map Representations

7. Example 2.1: Rotation by π/6 in x-y plane t : R2 → R2
Let

8. Let →

10. Example 2.2: → ∴
Let
Then

11. 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 H and H are matrix equivalent
if  nonsingular matrices P and Q s.t.
H = P H Q or H = P 1 H Q 1
Corollary 2.4:
Matrix equivalent matrices represent the same map, w.r.t. appropriate pair of bases.
Matrix equivalence classes.

12. 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 contain 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 mn matrix of rank k is matrix equivalent to the mn matrix that is all zeros except that the first k diagonal entries are ones.
Proof:
Gauss-Jordan reduction plus column reduction.

13. Example 2.7:
G-J row reduction:
Column reduction:
Column swapping:
Combined:

14. Corollary 2.8: 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 22 matrices have only three possible ranks: 0, 1, or 2.
Thus there are 3 matrix-equivalence classes.

15. If a linear map f : V n → W m is rank k,
then  some bases B → D s.t. f acts like a projection Rn → Rk  Rm.

16. Exercises 3.IV.2.
1. Show that, where A is a nonsingular square matrix, if P and Q are nonsingular square matrices such that PAQ = I then QP = A1 .
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?