![]() ![]() T = tB → B is diagonal, Then t and T are said to be diagonalizable. Find a basis D such thatħ Example 2.1: Rotation by π/6 in x-y plane t : R2 → R2ġ1 Consider t : V → V with matrix representation T w.r.t. Let p be a polynomial in P3 with where B = 1+x, 1x, x2+x3, x2x3 . Find the change of basis matrix for B, D R2. Corollary 1.5: A matrix is nonsingular it represents the identity map w.r.t. Hence, it equals to the product of elementary matrices, which can be shown to represent change of bases. ![]() Proof : (See Hefferon, p.239.) Nonsingular matrix is row equivalent to I. ![]() Proof : Bases changing matrix must be invertible, hence nonsingular. Lemma 1.2: Changing Basis Proof: Alternatively,Ĥ Lemma 1.4: A matrix changes bases iff it is nonsingular. Changing Representations of Vectorsĭefinition 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. Changing Representations of VectorsĢ 3.IV.1. ![]()
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