3.V. Change of Basis. 3.V.1. Changing Representations of Vectors 3.V.2. Changing Map Representations. 3.V.1. Changing Representations of Vectors. Definition 1.1 : Change of Basis Matrix
3.V.1. Changing Representations of Vectors
3.V.2. Changing Map Representations
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
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.
A matrix is nonsingular it represents the identity map w.r.t. some pair of bases.
1. Find the change of basis matrix for B, DR2.
(a) B= E2 , D= e2 , e1
(b) B= E2 ,
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
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
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
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.
Gauss-Jordan reduction plus column reduction.
Example 2.7 left-multiplication (:
G-J row 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.
Two same-sized matrices with the same rank are equivalent to the same block partial-identity matrix.
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.
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?
(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
(d) If two matrices are square and have matrix-equivalent squares, must they be matrix-equivalent?