Differential Equations and Linear Algebra (4th Edition)
4th Edition
ISBN: 9780321964670
Author: Stephen W. Goode, Scott A. Annin
Publisher: PEARSON
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Chapter 3.5, Problem 32AP
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ے ملزمة احمد
Q (a) Let f be a linear map from a space X into a space Y and (X1,X2,...,xn) basis for X, show that fis one-to-
one iff (f(x1),f(x2),...,f(x) } linearly independent.
(b) Let X= {ao+ax₁+a2x2+...+anxn, a;ER} be a vector space over R, write with prove a hyperspace and a
hyperplane of X.
مبر خد احمد
Q₂ (a) Let M be a subspace of a vector space X, and A= {fex/ f(x)=0, x E M ), show that whether A is
convex set or not, affine set or not.
Write with prove an
application of Hahn-Banach theorem.
Show that every singleton set in a normed space X is closed and any finite set in X is closed (14M)
Let M be a proper subspace of a finite dimension vector space X over a field F show that
whether: (1) If S is a base for M then S base for X or not, (2) If T base for X then base for M
or not.
(b) Let X-P₂(x) be a vector space over polynomials a field of real numbers R, write with L
prove convex subset of X and hyperspace of X.
Q₂/ (a) Let X-R³ be a vector space over a over a field of real numbers R and
A=((a,b,o), a,bE R), A is a subspace of X, let g be a function from A into R such that
gla,b,o)-a, gEA, find fe X such that g(t)=f(t), tEA.
(b) Let M be a non-empty subset of a space X, show that M is a hyperplane of X iff there
Xiff there
exists fE X/10) and tE F such that M=(xE X/ f(x)=t).
(c) Show that the relation equivalent is an equivalence relation on set of norms on a space
X.
Q/(a)Let X be a finite dimension vector space over a field F and S₁,S2CX such that S₁SS2. Show that
whether (1) if S, is a base for X then base for X or not (2) if S2 is a base for X then S, is a base for X or not
(b) Show that every subspace of vector space is convex and affine set but the conevrse need not to be true.
allet M be a non-empty subset of a vector space X over a field F and x,EX. Show that M is a
hyperspace iff xo+ M is a hyperplane and xo€ xo+M.
bState Hahn-Banach theorem and write with prove an application about it.
Show that every singleten subset and finite subset of a normed space is closed.
Oxfallet f he a function from a normad roace YI
Show tha ir continuour aty.GYiff
Chapter 3 Solutions
Differential Equations and Linear Algebra (4th Edition)
Ch. 3.1 - True-False Review For items (a)(j), decide if the...Ch. 3.1 - True-False Review For items (a)(j), decide if the...Ch. 3.1 - True-False Review For items a-j, decide if the...Ch. 3.1 - For Problems 1-6, determine the number of...Ch. 3.1 - For Problems 1-6, determine the number of...Ch. 3.1 - For Problems 1-6, determine the number of...Ch. 3.1 - For Problems 1-6, determine the number of...Ch. 3.1 - Prob. 5PCh. 3.1 - Prob. 6PCh. 3.1 - Use Definition 3.1.8 to derive the general...
Ch. 3.1 - Prob. 11PCh. 3.1 - For Problems 1215, determine the values of the...Ch. 3.1 - Prob. 13PCh. 3.1 - Prob. 14PCh. 3.1 - For Problems 1215, determine the values of the...Ch. 3.1 - Prob. 16PCh. 3.1 - For Problems 16-42, evaluate the determinant of...Ch. 3.1 - Prob. 18PCh. 3.1 - For Problems 16-42, evaluate the determinant of...Ch. 3.1 - Prob. 20PCh. 3.1 - Prob. 21PCh. 3.1 - Prob. 22PCh. 3.1 - For Problems 16-42, evaluate the determinant of...Ch. 3.1 - Prob. 24PCh. 3.1 - For Problems 16-42, evaluate the determinant of...Ch. 3.1 - Prob. 26PCh. 3.1 - For Problem 1642 evaluate the determinant of...Ch. 3.1 - For Problem 1642 evaluate the determinant of...Ch. 3.1 - Prob. 29PCh. 3.1 - Prob. 30PCh. 3.1 - For Problem 1642 evaluate the determinant of...Ch. 3.1 - For Problem 1642 evaluate the determinant of...Ch. 3.1 - For Problem 1642 evaluate the determinant of...Ch. 3.1 - For Problem 1642 evaluate the determinant of...Ch. 3.1 - Prob. 36PCh. 3.1 - Prob. 37PCh. 3.1 - Prob. 38PCh. 3.1 - For Problems 16-42, evaluate the determinant of...Ch. 3.1 - For Problems 16-42, evaluate the determinant of...Ch. 3.1 - For Problems 16-42, evaluate the determinant of...Ch. 3.1 - Prob. 42PCh. 3.1 - For Problems 43-46, evaluate the determinant of...Ch. 3.1 - For Problems 43-46, evaluate the determinant of...Ch. 3.1 - Prob. 45PCh. 3.1 - Prob. 46PCh. 3.1 - In Problem 4748, we explore a relationship between...Ch. 3.1 - Prob. 48PCh. 3.1 - (a) Write all 24 distinct permutations of the...Ch. 3.1 - Prob. 50PCh. 3.1 - Prob. 52PCh. 3.1 - 3.1Problems a) If A=[a11a12a21a22] and c is a...Ch. 3.1 - Prob. 55PCh. 3.1 - Prob. 56PCh. 3.1 - Let A be an arbitrary 44 matrix. By experimenting...Ch. 3.1 - Prob. 59PCh. 3.2 - For items a-f, decide if the given statement is...Ch. 3.2 - Prob. 2TFRCh. 3.2 - For items a-f, decide if the given statement is...Ch. 3.2 - Prob. 4TFRCh. 3.2 - Prob. 5TFRCh. 3.2 - Prob. 6TFRCh. 3.2 - For Problems 1-14, evaluate the determinant of the...Ch. 3.2 - For Problems 1-14, evaluate the determinant of the...Ch. 3.2 - For Problems 1-14, evaluate the determinant of the...Ch. 3.2 - For Problems 1-14, evaluate the determinant of the...Ch. 3.2 - For Problems 1-14, evaluate the determinant of the...Ch. 3.2 - For Problems 1-14, evaluate the determinant of the...Ch. 3.2 - For Problems 1-14, evaluate the determinant of the...Ch. 3.2 - Prob. 8PCh. 3.2 - For Problems 1-14, evaluate the determinant of the...Ch. 3.2 - Prob. 10PCh. 3.2 - For Problems 1-14, evaluate the determinant of the...Ch. 3.2 - Prob. 12PCh. 3.2 - For Problems 1-14, evaluate the determinant of the...Ch. 3.2 - Prob. 14PCh. 3.2 - For Problems 1521, use Theorem 3.2.5 to determine...Ch. 3.2 - Prob. 16PCh. 3.2 - For Problems 1521, use Theorem 3.2.5 to determine...Ch. 3.2 - Prob. 18PCh. 3.2 - Prob. 19PCh. 3.2 - Prob. 20PCh. 3.2 - For Problems 1521, use Theorem 3.2.5 to determine...Ch. 3.2 - Prob. 22PCh. 3.2 - Determine all values of the constant k for which...Ch. 3.2 - Determine all values of the constant k for which...Ch. 3.2 - Determine all values of the constant k for which...Ch. 3.2 - If A=[112314013], find det(A), and use properties...Ch. 3.2 - Prob. 27PCh. 3.2 - Verify property P9 for the matrices...Ch. 3.2 - For Problems 2932, let A=[abcd] and assume...Ch. 3.2 - For Problems 2932, let A=[abcd] and assume...Ch. 3.2 - Prob. 31PCh. 3.2 - Prob. 32PCh. 3.2 - For Problems 33-36, let A=[abcdefghi] and assume...Ch. 3.2 - For Problems 33-36, let A=[abcdefghi] and assume...Ch. 3.2 - For Problems 33-36, let A=[abcdefghi] and assume...Ch. 3.2 - Prob. 36PCh. 3.2 - For Problems 37-44, let A and B be 44 matrices...Ch. 3.2 - For Problems 37-44, let A and B be 44 matrices...Ch. 3.2 - For Problems 37-44, let A and B be 44 matrices...Ch. 3.2 - For Problems 37-44, let A and B be 44 matrices...Ch. 3.2 - For Problems 37-44, let A and B be 44 matrices...Ch. 3.2 - For Problems 37-44, let A and B be 44 matrices...Ch. 3.2 - For Problems 37-44, let A and B be 44 matrices...Ch. 3.2 - For Problems 37-44, let A and B be 44 matrices...Ch. 3.2 - Let A,B,andS be nn matrices. If S1AS=B, must A=B?...Ch. 3.2 - Let A=[124316k32]. a) In terms of k, find the...Ch. 3.2 - Without expanding the determinant, determine all...Ch. 3.2 - Use only properties P1,P2,P6 to show that...Ch. 3.2 - Prob. 49PCh. 3.2 - Prob. 50PCh. 3.2 - An nn matrix A that satisfies AT=A1 is called an...Ch. 3.2 - a. Use the definition of a determinant to prove...Ch. 3.2 - Use the determinants to prove that if A is...Ch. 3.2 - If A and S are nn matrices with S invertible, show...Ch. 3.2 - Prob. 55PCh. 3.2 - Let E be an elemetary matrix. Verify the formula...Ch. 3.2 - Show that |xy1x1y11x2y21|=0 represents the...Ch. 3.2 - Without expanding the determinant, show that...Ch. 3.2 - If A is an nn skew symmetric matrix and n is odd,...Ch. 3.2 - Prob. 60PCh. 3.2 - Let A be general 44 matrix. a Verify property P1...Ch. 3.2 - Prob. 62PCh. 3.2 - Determine all values of a for which...Ch. 3.2 - Prob. 64PCh. 3.2 - Prob. 65PCh. 3.3 - For items (a)(j), decide if the given statement is...Ch. 3.3 - For items (a)(j), decide if the given statement is...Ch. 3.3 - Prob. 5TFRCh. 3.3 - Prob. 6TFRCh. 3.3 - Prob. 7TFRCh. 3.3 - Prob. 8TFRCh. 3.3 - Prob. 9TFRCh. 3.3 - Prob. 1PCh. 3.3 - Prob. 2PCh. 3.3 - For Problems 1-4, determine all minors and...Ch. 3.3 - Prob. 4PCh. 3.3 - Prob. 5PCh. 3.3 - Prob. 6PCh. 3.3 - Prob. 7PCh. 3.3 - Prob. 8PCh. 3.3 - For Problems 7-14, use the cofactor expansion...Ch. 3.3 - Prob. 10PCh. 3.3 - Prob. 11PCh. 3.3 - For Problems 7-14, Use the cofactor expansion...Ch. 3.3 - Use the cofactor expansion theorem to evaluate the...Ch. 3.3 - Prob. 14PCh. 3.3 - For Problems 1522, evaluate the given determinant...Ch. 3.3 - Prob. 16PCh. 3.3 - Prob. 17PCh. 3.3 - Prob. 18PCh. 3.3 - Prob. 19PCh. 3.3 - For Problems 1522, evaluate the given determinant...Ch. 3.3 - For Problems 1522, evaluate the given determinant...Ch. 3.3 - Prob. 22PCh. 3.3 - Prob. 23PCh. 3.3 - Prob. 24PCh. 3.3 - Prob. 25PCh. 3.3 - Prob. 26PCh. 3.3 - Prob. 27PCh. 3.3 - Prob. 28PCh. 3.3 - Prob. 29PCh. 3.3 - Prob. 30PCh. 3.3 - Prob. 31PCh. 3.3 - For Problems 3138, determine the eigenvalues of...Ch. 3.3 - For Problems 3138, determine the eigenvalues of...Ch. 3.3 - For Problems 3138, determine the eigenvalues of...Ch. 3.3 - For Problems 3138, determine the eigenvalues of...Ch. 3.3 - For Problems 3138, determine the eigenvalues of...Ch. 3.3 - Prob. 37PCh. 3.3 - Prob. 39PCh. 3.3 - Prob. 40PCh. 3.3 - Prob. 41PCh. 3.3 - Prob. 42PCh. 3.3 - Prob. 43PCh. 3.3 - Prob. 44PCh. 3.3 - Prob. 45PCh. 3.3 - Prob. 46PCh. 3.3 - Prob. 47PCh. 3.3 - Prob. 48PCh. 3.3 - Prob. 49PCh. 3.3 - Prob. 50PCh. 3.3 - Prob. 51PCh. 3.3 - Prob. 52PCh. 3.3 - Prob. 53PCh. 3.3 - Prob. 54PCh. 3.3 - Prob. 55PCh. 3.3 - Prob. 57PCh. 3.3 - Prob. 58PCh. 3.3 - Prob. 59PCh. 3.3 - Prob. 60PCh. 3.3 - For Problems 59-64, use Cramers rule to solve the...Ch. 3.3 - Prob. 62PCh. 3.3 - Prob. 63PCh. 3.3 - Prob. 64PCh. 3.3 - Prob. 65PCh. 3.3 - Prob. 66PCh. 3.3 - Prob. 67PCh. 3.3 - Prob. 68PCh. 3.3 - Prob. 69PCh. 3.3 - Let A be a randomly generated invertible 44...Ch. 3.3 - Prob. 72PCh. 3.4 - For Problems 1-8, evaluate the given determinant....Ch. 3.4 - For Problem 1-8, evaluate the given determinant....Ch. 3.4 - For Problem 1-8, evaluate the given determinant....Ch. 3.4 - For Problem 1-8, evaluate the given determinant....Ch. 3.4 - For Problem 1-8, evaluate the given determinant....Ch. 3.4 - Prob. 6PCh. 3.4 - For Problem 1-8, evaluate the given determinant....Ch. 3.4 - For Problem 1-8, evaluate the given determinant....Ch. 3.4 - Prob. 9PCh. 3.4 - Prob. 10PCh. 3.4 - Prob. 11PCh. 3.4 - For problems 9-14, find det(A). If A is...Ch. 3.4 - Prob. 13PCh. 3.4 - For problems 9-14, find det(A). If A is...Ch. 3.4 - Prob. 15PCh. 3.4 - Prob. 16PCh. 3.4 - For Problems 15-20, use Cramers rule to determine...Ch. 3.4 - Prob. 18PCh. 3.4 - For Problems 15-20, use Cramers rule to determine...Ch. 3.4 - Prob. 20PCh. 3.4 - Prob. 21PCh. 3.4 - Prob. 22PCh. 3.4 - Prob. 23PCh. 3.4 - For Problems 23-29, assume that A and B be 33...Ch. 3.4 - Prob. 25PCh. 3.4 - Prob. 26PCh. 3.4 - For Problems 23-29, assume that A and B be 33...Ch. 3.4 - For Problems 23-29, assume that A and B be 33...Ch. 3.4 - Prob. 29PCh. 3.5 - For Problems 1-6, evaluate the determinant of the...Ch. 3.5 - For Problems 1-6, evaluate the determinant of the...Ch. 3.5 - For Problems 1-6, evaluate the determinant of the...Ch. 3.5 - For Problems 1-6, evaluate the determinant of the...Ch. 3.5 - For Problems 16, evaluate the determinant of the...Ch. 3.5 - Prob. 6APCh. 3.5 - Prob. 7APCh. 3.5 - Prob. 8APCh. 3.5 - Prob. 9APCh. 3.5 - Prob. 10APCh. 3.5 - For Problem 11-14, suppose A and B are 44...Ch. 3.5 - For Problem 11-14, suppose A and B are 44...Ch. 3.5 - Prob. 13APCh. 3.5 - Prob. 14APCh. 3.5 - Prob. 15APCh. 3.5 - Prob. 16APCh. 3.5 - Prob. 17APCh. 3.5 - Prob. 18APCh. 3.5 - Prob. 19APCh. 3.5 - Prob. 20APCh. 3.5 - Prob. 21APCh. 3.5 - Prob. 22APCh. 3.5 - Prob. 23APCh. 3.5 - Prob. 24APCh. 3.5 - Prob. 25APCh. 3.5 - Prob. 26APCh. 3.5 - Prob. 27APCh. 3.5 - Prob. 28APCh. 3.5 - Prob. 29APCh. 3.5 - Prob. 30APCh. 3.5 - Prob. 31APCh. 3.5 - Prob. 32APCh. 3.5 - Prob. 33APCh. 3.5 - True or false: Given any real number r and any 33...Ch. 3.5 - Prob. 35APCh. 3.5 - Prob. 36APCh. 3.5 - Prob. 37APCh. 3.5 - Let A and B be nn matrices such that AB=BA. Use...Ch. 3.5 - A real nn matrix A is called orthogonal if...Ch. 3.5 - For Problems 40-42, Use Cramers rule to solve the...Ch. 3.5 - For Problems 4042, use Cramers rule to solve the...Ch. 3.5 - Prob. 42AP
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