a) Let v₁ = V2 = and denote E = , {V1, V2}. Suppose that the coordi- nates of w in the basis ε is [w] = Determine w. b) Find [P]B - the coordinates of vector p(x) =5x+5 with respect to the basis B = {x, 1+x} of P₁, the real vector space of polynomials in x with real coefficients and degree at most 1. c) Let v₁ = = (+) V2= (2) coordinates [w]ε. and denote E = {V1, V2}. For w = a) Find the characteristic polynomial of the following matrix C 0 0 0 C = 0 1 -2 0 -2 4 b) Find the eigenvalues of the matrix A = 2 3 3 2 (3) € R² find the c) For each eigenvalue of A in Part (b), find a corresponding eigenvector. d) Based on your answer in Part (b), classify the matrix A as positive-definite, negative- definite, or indefinite. Justify your answer. a) Consider the quadratic form Q(x, y) = 2x²+3y²+2xy. Find A such that Q(x, y) = x Ax, where x =
a) Let v₁ = V2 = and denote E = , {V1, V2}. Suppose that the coordi- nates of w in the basis ε is [w] = Determine w. b) Find [P]B - the coordinates of vector p(x) =5x+5 with respect to the basis B = {x, 1+x} of P₁, the real vector space of polynomials in x with real coefficients and degree at most 1. c) Let v₁ = = (+) V2= (2) coordinates [w]ε. and denote E = {V1, V2}. For w = a) Find the characteristic polynomial of the following matrix C 0 0 0 C = 0 1 -2 0 -2 4 b) Find the eigenvalues of the matrix A = 2 3 3 2 (3) € R² find the c) For each eigenvalue of A in Part (b), find a corresponding eigenvector. d) Based on your answer in Part (b), classify the matrix A as positive-definite, negative- definite, or indefinite. Justify your answer. a) Consider the quadratic form Q(x, y) = 2x²+3y²+2xy. Find A such that Q(x, y) = x Ax, where x =
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.5: Basis And Dimension
Problem 69E: Find a basis for R2 that includes the vector (2,2).
Related questions
Question
![a) Let v₁ =
V2 =
and denote E
=
,
{V1, V2}. Suppose that the coordi-
nates of w in the basis ε is [w] =
Determine w.
b) Find [P]B - the coordinates of vector p(x) =5x+5 with respect to the basis
B = {x, 1+x} of P₁, the real vector space of polynomials in x with real coefficients
and degree at most 1.
c) Let v₁ =
= (+)
V2=
(2)
coordinates [w]ε.
and denote E = {V1, V2}. For w =
a) Find the characteristic polynomial of the following matrix C
0 0 0
C = 0 1 -2
0
-2 4
b) Find the eigenvalues of the matrix
A =
2 3
3 2
(3)
€ R² find the
c) For each eigenvalue of A in Part (b), find a corresponding eigenvector.
d) Based on your answer in Part (b), classify the matrix A as positive-definite, negative-
definite, or indefinite. Justify your answer.
a) Consider the quadratic form Q(x, y) = 2x²+3y²+2xy. Find A such that Q(x, y) =
x Ax, where x =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb15d15d9-e398-4283-aed7-d6fa30a19f66%2F8385c2ce-da9d-4b61-8d2c-0566b80d8761%2Fi1h9tkn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:a) Let v₁ =
V2 =
and denote E
=
,
{V1, V2}. Suppose that the coordi-
nates of w in the basis ε is [w] =
Determine w.
b) Find [P]B - the coordinates of vector p(x) =5x+5 with respect to the basis
B = {x, 1+x} of P₁, the real vector space of polynomials in x with real coefficients
and degree at most 1.
c) Let v₁ =
= (+)
V2=
(2)
coordinates [w]ε.
and denote E = {V1, V2}. For w =
a) Find the characteristic polynomial of the following matrix C
0 0 0
C = 0 1 -2
0
-2 4
b) Find the eigenvalues of the matrix
A =
2 3
3 2
(3)
€ R² find the
c) For each eigenvalue of A in Part (b), find a corresponding eigenvector.
d) Based on your answer in Part (b), classify the matrix A as positive-definite, negative-
definite, or indefinite. Justify your answer.
a) Consider the quadratic form Q(x, y) = 2x²+3y²+2xy. Find A such that Q(x, y) =
x Ax, where x =
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