Linear Algebra: A Modern Introduction
4th Edition
ISBN: 9781285463247
Author: David Poole
Publisher: Cengage Learning
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Chapter 1, Problem 16RQ
To determine
To find:
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Let
2
A =
4
3
-4
0
1
(a) Show that v =
eigenvalue.
()
is an eigenvector of A and find the corresponding
(b) Find the characteristic polynomial of A and factorise it. Hint: the answer to (a)
may be useful.
(c) Determine all eigenvalues of A and find bases for the corresponding eigenspaces.
(d) Find an invertible matrix P and a diagonal matrix D such that P-¹AP = D.
(c) Let
6
0 0
A =
-10 4 8
5 1 2
(i) Find the characteristic polynomial of A and factorise it.
(ii) Determine all eigenvalues of A and find bases for the corresponding
eigenspaces.
(iii) Is A diagonalisable? Give reasons for your answer.
most 2, and let
Let P2 denote the vector space of polynomials of degree at
D: P2➡ P2
be the transformation that sends a polynomial p(t) = at² + bt+c in P2 to its derivative
p'(t)
2at+b, that is,
D(p) = p'.
(a) Prove that D is a linear transformation.
(b) Find a basis for the kernel ker(D) of the linear transformation D and compute its
nullity.
(c) Find a basis for the image im(D) of the linear transformation D and compute its
rank.
(d) Verify that the Rank-Nullity Theorem holds for the linear transformation D.
(e) Find the matrix representation of D in the standard basis (1,t, t2) of P2.
Chapter 1 Solutions
Linear Algebra: A Modern Introduction
Ch. 1.1 - Draw the following vectors in standard position in...Ch. 1.1 - Prob. 2EQCh. 1.1 - Prob. 3EQCh. 1.1 - For each of the following pairs of points, draw...Ch. 1.1 - Prob. 12EQCh. 1.1 - In Figure 1.24, A, B, C, D, E, and F are the...Ch. 1.1 - In Exercises 15 and 16, simplify the given vector...Ch. 1.1 - In Exercises 15 and 16, simplify the given vector...Ch. 1.1 - In Exercises 17 and 18, solve for the vector x in...Ch. 1.1 - In Exercises 17 and 18, solve for the vector x in...
Ch. 1.1 - In Exercises 19 and 20, draw the coordinate axes...Ch. 1.1 - In Exercises 21 and 22, draw the standard...Ch. 1.1 - In Exercises 25-28, u and v are binary vectors....Ch. 1.1 - In Exercises 25-28, u and v are binary vectors....Ch. 1.1 - In Exercises 25-28, u and v are binary vectors....Ch. 1.1 - In Exercises 25-28, u and v are binary vectors....Ch. 1.1 - Write out the addition and multiplication tables...Ch. 1.1 - Write out the addition and multiplication tables...Ch. 1.1 - In Exercises 31-43, perform the indicated...Ch. 1.1 - In Exercises 31-43, perform the indicated...Ch. 1.1 - In Exercises 31-43, perform the indicated...Ch. 1.1 - In Exercises 31-43, perform the indicated...Ch. 1.1 - In Exercises 31-43, perform the indicated...Ch. 1.1 - In Exercises 31-43, perform the indicated...Ch. 1.1 - In Exercises 31-43, perform the indicated...Ch. 1.1 - In Exercises 31-43, perform the indicated...Ch. 1.1 - Prob. 39EQCh. 1.1 - Prob. 40EQCh. 1.1 - In Exercises 31-43, perform the indicated...Ch. 1.1 - In Exercises 31-43, perform the indicated...Ch. 1.1 - In Exercises 31-43, perform the indicated...Ch. 1.1 - In Exercises 44-55, solve the given equation or...Ch. 1.1 - In Exercises 44-55, solve the given equation or...Ch. 1.1 - In Exercises 44-55, solve the given equation or...Ch. 1.1 - In Exercises 44-55, solve the given equation or...Ch. 1.1 - In Exercises 44-55, solve the given equation or...Ch. 1.1 - In Exercises 44-55, solve the given equation or...Ch. 1.1 - In Exercises 44-55, solve the given equation or...Ch. 1.1 - Prob. 51EQCh. 1.1 - In Exercises 44-55, solve the given equation or...Ch. 1.1 - In Exercises 44-55, solve the given equation or...Ch. 1.1 - Prob. 54EQCh. 1.1 - In Exercises 44-55, solve the given equation or...Ch. 1.2 - In Exercises 1-6, find .
1.
Ch. 1.2 - In Exercises 1-6, find .
2.
Ch. 1.2 - In Exercises 1-6, find uv. u=[123],v=[231]Ch. 1.2 - In Exercises 1-6, find uv....Ch. 1.2 - In Exercises 13-16, find the distance...Ch. 1.2 - In Exercises 1-6, find .
6.
Ch. 1.2 - In Exercises 7-12, find for the given exercise,...Ch. 1.2 - In Exercises 7-12, find u for the given exercise,...Ch. 1.2 - In Exercises 7-12, find for the given exercise,...Ch. 1.2 - In Exercises 7-12, find u for the given exercise,...Ch. 1.2 - In Exercises 7-12, find for the given exercise,...Ch. 1.2 - In Exercises 7-12, find u for the given exercise,...Ch. 1.2 - In Exercises 13-16, find the distance between and...Ch. 1.2 - In Exercises 13-16, find the distance between and...Ch. 1.2 - Prob. 15EQCh. 1.2 - Prob. 16EQCh. 1.2 - In Exercises 18-23, determine whether the angle...Ch. 1.2 - In Exercises 18-23, determine whether the angle...Ch. 1.2 - In Exercises 18-23, determine whether the angle...Ch. 1.2 - In Exercises 18-23, determine whether the angle...Ch. 1.2 - In Exercises 18-23, determine whether the angle...Ch. 1.2 - Prob. 23EQCh. 1.2 - Prob. 24EQCh. 1.2 - Prob. 25EQCh. 1.2 - Prob. 26EQCh. 1.2 - Prob. 27EQCh. 1.2 - Prob. 28EQCh. 1.2 - Prob. 29EQCh. 1.2 -
In Exercises 40-45, find the projection of v onto...Ch. 1.2 - In Exercises 40-45, find the projection of vontou....Ch. 1.2 - Prob. 44EQCh. 1.2 - Prob. 45EQCh. 1.2 - In Exercises 48 and 49, find all values of the...Ch. 1.2 - In Exercises 48 and 49, find all values of the...Ch. 1.2 - Describe all vectors v=[xy] that are orthogonal to...Ch. 1.3 - In Exercises 1 and 2, write the equation of the...Ch. 1.3 - In Exercises 1 and 2, write the equation of the...Ch. 1.3 - Prob. 3EQCh. 1.3 - Prob. 4EQCh. 1.3 - Prob. 5EQCh. 1.3 - In Exercises 3-6, write the equation of the line...Ch. 1.3 - Prob. 7EQCh. 1.3 - In Exercises 7 and 8, write the equation of the...Ch. 1.3 - Prob. 9EQCh. 1.3 - In Exercises 9 and 10, write the equation of the...Ch. 1.3 - Prob. 11EQCh. 1.3 - In Exercises 11 and 12, give the vector equation...Ch. 1.3 - In Exercises 13 and 14, give the vector equation...Ch. 1.3 - In Exercises 13 and 14, give the vector equation...Ch. 1.3 - Find parametric equations and an equation in...Ch. 1.3 - Prob. 18EQCh. 1.3 - Prob. 19EQCh. 1.3 - 20. Find the vector form of the equation of the...Ch. 1.3 - Find the vector form of the equation of the line...Ch. 1.3 - Find the vector form of the equation of the line...Ch. 1.3 - Prob. 23EQCh. 1.3 - 24. Find the normal form of the equation of the...Ch. 1.3 - 26. Find the equation of the set of all points...Ch. 1.3 - In Exercises 27 and 28, find the distance from the...Ch. 1.3 - In Exercises 29 and 30, find the distance from the...Ch. 1.3 - Prob. 30EQCh. 1.3 - In Exercises 35 and 36, find the distance between...Ch. 1.3 - Prob. 37EQCh. 1.3 - In Exercises 37 and 38, find the distance between...Ch. 1.3 - In Exercises 43-44, find the acute angle between...Ch. 1.3 - Prob. 44EQCh. 1.4 - A sign hanging outside Joes Diner has a mass of 50...Ch. 1 - Mark each of the following statements true or...Ch. 1 - 2. If , and the vector is drawn with its tail at...Ch. 1 - 3. If , and , solve for x.
Ch. 1 - Prob. 5RQCh. 1 - 6. Find the projection of .
Ch. 1 - 7. Find a unit vector in the xy-plane that is...Ch. 1 - 8. Find the general equation of the plane through...Ch. 1 - Find the general equation of the plane through the...Ch. 1 - 10. Find the general equation of the plane through...Ch. 1 - 12. Find the midpoint of the line segment...Ch. 1 - Prob. 13RQCh. 1 - 14. Find the distance from the point to the plane...Ch. 1 - Find the distance from the point (3,2,5) to the...Ch. 1 - Prob. 16RQCh. 1 - Prob. 17RQCh. 1 - 18. If possible, solve .
Ch. 1 - Prob. 19RQ
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