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

2. 3.V.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.V.1. 1. Find the change of basis matrix for B, D  R 2. (a) B = E 2, D =  e 2, e 1  (b) B = E 2, D = E 2 (c)(d) 2. Let p be a polynomial in P 3 with where B =  1+x, 1  x, x 2 +x 3, x 2  x 3 . Find a basis D such that

6. 3.V.2. Changing Map Representations

7. Example 2.1:Rotation by π/6 in x-y plane t : R 2 → R 2 Let

8. →

9. Example 2.2: → ∴ Let Then

10. Consider t : V → V with matrix representation T w.r.t. some basis. If  basis B s.t. T = t B → 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 orH = 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.

11. 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: andare 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.

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

13. 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.

14. If a linear map f : V n → W m is rank k, then  some bases B → D s.t. f acts like a projection R n → R m.

15. Exercises 3.V.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?