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Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.9: Properties Of Determinants

Problem 34E

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3. Let the matrix of L: R³ →R² with respect to the bases S =
and
T-{]-[-]}
T=
W₁ W₂
a. Find [L(v₁)T[L(V₂)]T [L(V3)]T
b. Find L(v₁), L(V₂), L(V3)
c. Find L
12
--CO
be A=
(
(100)

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9. Let and B 3 = {[B]-[-2]} = B' ={[2¹].[6]} be two ordered bases for R². Find the change of basis matrix [I]B'.
1 - [5] and v₂ - [3]. = Let V₁ = [T(v₁)]B = ( be the matrix for T:R² R2 with respect to the basis B = {V₁, V₂}. Find [T(v1)] and [T(v₂)]B. [T(v₂)]B = ( i i and let → i 4 -1 ^-43 A = tel )
Find the matrix A' for 7 relative to the basis 8¹. A' = T: R² R², T(x, y) = (x − y, 4y − x), B' = {(1, −2), (0, 3)) ↓ T
(4) Consider L: R2 R2 given by Consider the two bases S = [e₁, e2] and F = [u₁, u2], where (¹). (1¹). x1 2x1 L - (^₂) = (2,2²4 ¹₂). x2 + U₁ = We find that: (a) The representation matrix A of L with respect to S is 2 A = ([L(en)]s_[L(e2)]s) = (ii) 9). 1 (b) The representation matrix B of L with respect to F is and its inverse is 2 B = [L(U₂)]F) · ([L(ur)]F_ [L(us)]x) = (1 7¹). 0 U2 = (c) The transition matrix T from F to S is T = ([m]s [1₂]s) = (11¹), B = T-¹ AT = T-1 We can verify directly that 1/ = 1 (41) 2 2 2 - 1 (1 1) (² 9) ( 1 ) = (²1¹). 2 1 0
Consider B={1,x,x^2} and B'={1+x,1-x^2,2} Find the change of basis matrix 1.P from Bto B' 2.Q from B' to B Further find the relation between P and Q
Let 8 = {(1, 3), (-2,-2)} and 8 = {(-12, 0), (-4, 4)) be bases for R2, and let 4 2 A = - [83] 03 be the matrix for T: R2 R² relative to 8. (a) Find the transition matrix P from 8' to 8. P= ↓↑ (b) Use the matrices P and A to find [v] and [7(v)]g, where [v] = [4 -1]7. [7(v)]B (c) Find P-¹ and A' (the matrix for 7 relative to 8'). p-1= ↓1 -88- A'= ↓↑ (d) Find [7(v)]g' two ways. [7(v)] = P¹[7(v)]g = [7(v)]g¹ = A'[v]g' = = [v]8 = = ↓↑ 8 ↓ ↑
Advanced Math Question
Consider the quadratic form Q(x, y) = -² A = la cla 14 9 ala 10 Ila 28 9 ---/- 14 9 14 9 14 9 20 9 56 x y 9 following principal minors of the matrix A: The (leading) principal minor 1pm3 = pm3 of order 3: The leading principal minor 1pm² = pm3 of order 2: The leading principal minor 1pm₁ = pm of order 1: The principal minor pm² of order 2: The principal minor pm² of order 2: The principal minor pm of order 1: The principal minor pm of order 1: 9 Now use your answers to classify Q: (No answer given) -28 x z We want to classify this using Sylvester's Criteria (the principal minor test). Calculate the 28 y z 9 + 202² with symmetric coefficient matrix
7. Let be the operator on P[r]3 defined by L(p(x)) = xp'(x) +p"(x) (1) Find the matrix A representing with respect to [1, x, x²]. (2) Find the matrix B representing with respect to [1, x, 1 + x²]. (3) Find the matrix S such that B S-¹ AS. (4) If p(x) = ao + a₁x + a₂(1 + x²), = calculate L¹ (p(x)).
5. Let B = {(1,0), (0,1)} and B' = {(2, 2), (1,4)} be ordered bases for R² and let v = (−3, 10). a.) Find [v]B, the coordinate vector of v relative to B. b.) Find PB→B', the transition matrix from B to B'.

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