Linear Algebra With Applications (classic Version)
5th Edition
ISBN: 9780135162972
Author: BRETSCHER, OTTO
Publisher: Pearson Education, Inc.,
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Chapter 8.2, Problem 41E
To determine
To find:The dimension of the space
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Q.1) Classify the following statements as a true or false statements:
a. If M is a module, then every proper submodule of M is contained in a maximal
submodule of M.
b. The sum of a finite family of small submodules of a module M is small in M.
c. Zz is directly indecomposable.
d. An epimorphism a: M→ N is called solit iff Ker(a) is a direct summand in M.
e. The Z-module has two composition series.
Z
6Z
f. Zz does not have a composition series.
g. Any finitely generated module is a free module.
h. If O→A MW→ 0 is short exact sequence then f is epimorphism.
i. If f is a homomorphism then f-1 is also a homomorphism.
Maximal C≤A if and only if is simple.
Sup
Q.4) Give an example and explain your claim in each case:
Monomorphism not split.
b) A finite free module.
c) Semisimple module.
d) A small submodule A of a module N and a homomorphism op: MN, but
(A) is not small in M.
I need diagram with solutions
Chapter 8 Solutions
Linear Algebra With Applications (classic Version)
Ch. 8.1 - For each of the matrices in Exercises 1 through 6,...Ch. 8.1 - For each of the matrices in Exercises 1 through 6,...Ch. 8.1 - For each of the matrices in Exercises 1 through 6,...Ch. 8.1 - For each of the matrices in Exercises 1 through 6,...Ch. 8.1 - For each of the matrices in Exercises 1 through 6,...Ch. 8.1 - For each of the matrices in Exercises 1 through 6,...Ch. 8.1 - For each of the matrices A in Exercises 7 through...Ch. 8.1 - For each of the matrices A in Exercises 7 through...Ch. 8.1 - For each of the matrices A in Exercises 7 through...Ch. 8.1 - For each of the matrices A in Exercises 7 through...
Ch. 8.1 - For each of the matrices A in Exercises 7 through...Ch. 8.1 - Let L from R3 to R3 be the reflection about the...Ch. 8.1 - Consider a symmetric 33 matrix A with A2=I3 . Is...Ch. 8.1 - In Example 3 of this section, we diagonalized the...Ch. 8.1 - If A is invertible and orthogonally...Ch. 8.1 - Find the eigenvalues of the matrix...Ch. 8.1 - Use the approach of Exercise 16 to find the...Ch. 8.1 - Consider unit vector v1,...,vn in Rn such that the...Ch. 8.1 - Consider a linear transformation L from Rm to Rn ....Ch. 8.1 - Consider a linear transformation T from Rm to Rn ,...Ch. 8.1 - Consider a symmetric 33 matrix A with eigenvalues...Ch. 8.1 - Consider the matrix A=[0200k0200k0200k0] , where k...Ch. 8.1 - If an nn matrix A is both symmetric and...Ch. 8.1 - Consider the matrix A=[0001001001001000] . Find an...Ch. 8.1 - Consider the matrix [0000100010001000100010000] ....Ch. 8.1 - Let Jn be the nn matrix with all ones on the...Ch. 8.1 - Diagonalize the nn matrix (All ones along both...Ch. 8.1 - Diagonalize the 1313 matrix (All ones in the last...Ch. 8.1 - Consider a symmetric matrix A. If the vector v is...Ch. 8.1 - Consider an orthogonal matrix R whose first column...Ch. 8.1 - True or false? If A is a symmetric matrix, then...Ch. 8.1 - Consider the nn matrix with all ones on the main...Ch. 8.1 - For which angles(s) can you find three distinct...Ch. 8.1 - For which angles(s) can you find four distinct...Ch. 8.1 - Consider n+1 distinct unit vectors in Rn such that...Ch. 8.1 - Consider a symmetric nn matrix A with A2=A . Is...Ch. 8.1 - If A is any symmetric 22 matrix with eigenvalues...Ch. 8.1 - If A is any symmetric 22 matrix with eigenvalues...Ch. 8.1 - If A is any symmetric 33 matrix with eigenvalues...Ch. 8.1 - If A is any symmetric 33 matrix with eigenvalues...Ch. 8.1 - Show that for every symmetric nn matrix A, there...Ch. 8.1 - Find a symmetric 22 matrix B such that...Ch. 8.1 - For A=[ 2 11 11 11 2 11 11 11 2 ] find a nonzero...Ch. 8.1 - Consider an invertible symmetric nn matrix A. When...Ch. 8.1 - We say that an nnmatrix A is triangulizable if A...Ch. 8.1 - a. Consider a complex upper triangular nnmatrix U...Ch. 8.1 - Let us first introduce two notations. For a...Ch. 8.1 - Let U0 be a real upper triangular nn matrix with...Ch. 8.1 - Let R be a complex upper triangular nnmatrix with...Ch. 8.1 - Let A be a complex nnmatrix that ||1 for all...Ch. 8.2 - For each of the quadratic forms q listed in...Ch. 8.2 - For each of the quadratic forms q listed in...Ch. 8.2 - For each of the quadratic forms q listed in...Ch. 8.2 - Determine the definiteness of the quadratic forms...Ch. 8.2 - Determine the definiteness of the quadratic forms...Ch. 8.2 - Determine the definiteness of the quadratic forms...Ch. 8.2 - Determine the definiteness of the quadratic forms...Ch. 8.2 - If A is a symmetric matrix, what can you say about...Ch. 8.2 - Recall that a real square matrix A is called skew...Ch. 8.2 - Consider a quadratic form q(x)=xAx on n and a...Ch. 8.2 - If A is an invertible symmetric matrix, what is...Ch. 8.2 - Show that a quadratic form q(x)=xAx of two...Ch. 8.2 - Show that the diagonal elements of a positive...Ch. 8.2 - Consider a 22 matrix A=[abbc] , where a and det A...Ch. 8.2 - Sketch the curves defined in Exercises 15 through...Ch. 8.2 - Sketch the curves defined in Exercises 15 through...Ch. 8.2 - Sketch the curves defined in Exercises 15 through...Ch. 8.2 - Sketch the curves defined in Exercises 15 through...Ch. 8.2 - Sketch the curves defined in Exercises 15 through...Ch. 8.2 - Sketch the curves defined in Exercises 15 through...Ch. 8.2 - a. Sketch the following three surfaces:...Ch. 8.2 - On the surface x12+x22x32+10x1x3=1 find the two...Ch. 8.2 - Prob. 23ECh. 8.2 - Consider a quadratic form q(x)=xAx Where A is a...Ch. 8.2 - Prob. 25ECh. 8.2 - Prob. 26ECh. 8.2 - Consider a quadratic form q(x)=xAx , where A is a...Ch. 8.2 - Show that any positive definite nnmatrix A can be...Ch. 8.2 - For the matrix A=[8225] , write A=BBT as discussed...Ch. 8.2 - Show that any positive definite matrix A can be...Ch. 8.2 - Prob. 31ECh. 8.2 - Prob. 32ECh. 8.2 - Prob. 33ECh. 8.2 - Prob. 34ECh. 8.2 - Prob. 35ECh. 8.2 - Prob. 36ECh. 8.2 - Prob. 37ECh. 8.2 - Prob. 38ECh. 8.2 - Prob. 39ECh. 8.2 - Prob. 40ECh. 8.2 - Prob. 41ECh. 8.2 - Prob. 42ECh. 8.2 - Prob. 43ECh. 8.2 - Prob. 44ECh. 8.2 - Prob. 45ECh. 8.2 - Prob. 46ECh. 8.2 - Prob. 47ECh. 8.2 - Prob. 48ECh. 8.2 - Prob. 49ECh. 8.2 - Prob. 50ECh. 8.2 - What are the signs of the determinants of the...Ch. 8.2 - Consider a quadratic form q. If A is a symmetric...Ch. 8.2 - Consider a quadratic form q(x1,...,xn) with...Ch. 8.2 - If A is a positive semidefinite matrix with a11=0...Ch. 8.2 - Prob. 55ECh. 8.2 - Prob. 56ECh. 8.2 - Prob. 57ECh. 8.2 - Prob. 58ECh. 8.2 - Prob. 59ECh. 8.2 - Prob. 60ECh. 8.2 - Prob. 61ECh. 8.2 - Prob. 62ECh. 8.2 - Prob. 63ECh. 8.2 - Prob. 64ECh. 8.2 - Prob. 65ECh. 8.2 - Prob. 66ECh. 8.2 - Prob. 67ECh. 8.2 - Prob. 68ECh. 8.2 - Prob. 69ECh. 8.2 - Prob. 70ECh. 8.2 - Prob. 71ECh. 8.3 - Find the singular values of A=[1002] .Ch. 8.3 - Let A be an orthogonal 22 matrix. Use the image of...Ch. 8.3 - Let A be an orthogonal nn matrix. Find the...Ch. 8.3 - Find the singular values of A=[1101] .Ch. 8.3 - Find the singular values of A=[pqqp] . Explain...Ch. 8.3 - Prob. 6ECh. 8.3 - Prob. 7ECh. 8.3 - Find singular value decompositions for the...Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - If A is an invertible 22 matrix, what is the...Ch. 8.3 - If A is an invertible nn matrix, what is the...Ch. 8.3 - Consider an nm matrix A with rank(A)=m , and a...Ch. 8.3 - Prob. 18ECh. 8.3 - Prob. 19ECh. 8.3 - Prob. 20ECh. 8.3 - Prob. 21ECh. 8.3 - Consider the standard matrix A representing the...Ch. 8.3 - Consider an SVD A=UVT of an nm matrix A. Show that...Ch. 8.3 - If A is a symmetric nn matrix, what is the...Ch. 8.3 - Prob. 25ECh. 8.3 - Prob. 26ECh. 8.3 - Prob. 27ECh. 8.3 - Prob. 28ECh. 8.3 - Prob. 29ECh. 8.3 - Prob. 30ECh. 8.3 - Show that any matrix of rank r can be written as...Ch. 8.3 - Prob. 32ECh. 8.3 - Prob. 33ECh. 8.3 - For which square matrices A is there a singular...Ch. 8.3 - Prob. 35ECh. 8.3 - Prob. 36ECh. 8 - The singular values of any diagonal matrix D are...Ch. 8 - Prob. 2ECh. 8 - Prob. 3ECh. 8 - Prob. 4ECh. 8 - Prob. 5ECh. 8 - Prob. 6ECh. 8 - The function q(x1,x2)=3x12+4x1x2+5x2 is a...Ch. 8 - Prob. 8ECh. 8 - If matrix A is positive definite, then all the...Ch. 8 - Prob. 10ECh. 8 - Prob. 11ECh. 8 - Prob. 12ECh. 8 - Prob. 13ECh. 8 - Prob. 14ECh. 8 - Prob. 15ECh. 8 - Prob. 16ECh. 8 - Prob. 17ECh. 8 - Prob. 18ECh. 8 - Prob. 19ECh. 8 - Prob. 20ECh. 8 - Prob. 21ECh. 8 - Prob. 22ECh. 8 - If A and S are invertible nn matrices, then...Ch. 8 - Prob. 24ECh. 8 - Prob. 25ECh. 8 - Prob. 26ECh. 8 - Prob. 27ECh. 8 - Prob. 28ECh. 8 - Prob. 29ECh. 8 - Prob. 30ECh. 8 - Prob. 31ECh. 8 - Prob. 32ECh. 8 - Prob. 33ECh. 8 - Prob. 34ECh. 8 - Prob. 35ECh. 8 - Prob. 36ECh. 8 - Prob. 37ECh. 8 - Prob. 38ECh. 8 - Prob. 39ECh. 8 - Prob. 40ECh. 8 - Prob. 41ECh. 8 - Prob. 42ECh. 8 - Prob. 43ECh. 8 - Prob. 44ECh. 8 - Prob. 45ECh. 8 - Prob. 46ECh. 8 - Prob. 47ECh. 8 - Prob. 48ECh. 8 - Prob. 49ECh. 8 - Prob. 50ECh. 8 - Prob. 51ECh. 8 - Prob. 52ECh. 8 - Prob. 53ECh. 8 - Prob. 54E
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- T. Determine the least common denominator and the domain for the 2x-3 10 problem: + x²+6x+8 x²+x-12 3 2x 2. Add: + Simplify and 5x+10 x²-2x-8 state the domain. 7 3. Add/Subtract: x+2 1 + x+6 2x+2 4 Simplify and state the domain. x+1 4 4. Subtract: - Simplify 3x-3 x²-3x+2 and state the domain. 1 15 3x-5 5. Add/Subtract: + 2 2x-14 x²-7x Simplify and state the domain.arrow_forwardQ.1) Classify the following statements as a true or false statements: Q a. A simple ring R is simple as a right R-module. b. Every ideal of ZZ is small ideal. very den to is lovaginz c. A nontrivial direct summand of a module cannot be large or small submodule. d. The sum of a finite family of small submodules of a module M is small in M. e. The direct product of a finite family of projective modules is projective f. The sum of a finite family of large submodules of a module M is large in M. g. Zz contains no minimal submodules. h. Qz has no minimal and no maximal submodules. i. Every divisible Z-module is injective. j. Every projective module is a free module. a homomorp cements Q.4) Give an example and explain your claim in each case: a) A module M which has a largest proper submodule, is directly indecomposable. b) A free subset of a module. c) A finite free module. d) A module contains no a direct summand. e) A short split exact sequence of modules.arrow_forwardListen ANALYZING RELATIONSHIPS Describe the x-values for which (a) f is increasing or decreasing, (b) f(x) > 0 and (c) f(x) <0. y Af -2 1 2 4x a. The function is increasing when and decreasing whenarrow_forwardBy forming the augmented matrix corresponding to this system of equations and usingGaussian elimination, find the values of t and u that imply the system:(i) is inconsistent.(ii) has infinitely many solutions.(iii) has a unique solutiona=2 b=1arrow_forwardif a=2 and b=1 1) Calculate 49(B-1)2+7B−1AT+7ATB−1+(AT)2 2)Find a matrix C such that (B − 2C)-1=A 3) Find a non-diagonal matrix E ̸= B such that det(AB) = det(AE)arrow_forwardWrite the equation line shown on the graph in slope, intercept form.arrow_forward1.2.15. (!) Let W be a closed walk of length at least 1 that does not contain a cycle. Prove that some edge of W repeats immediately (once in each direction).arrow_forward1.2.18. (!) Let G be the graph whose vertex set is the set of k-tuples with elements in (0, 1), with x adjacent to y if x and y differ in exactly two positions. Determine the number of components of G.arrow_forward1.2.17. (!) Let G,, be the graph whose vertices are the permutations of (1,..., n}, with two permutations a₁, ..., a,, and b₁, ..., b, adjacent if they differ by interchanging a pair of adjacent entries (G3 shown below). Prove that G,, is connected. 132 123 213 312 321 231arrow_forward1.2.19. Let and s be natural numbers. Let G be the simple graph with vertex set Vo... V„−1 such that v; ↔ v; if and only if |ji| Є (r,s). Prove that S has exactly k components, where k is the greatest common divisor of {n, r,s}.arrow_forward1.2.20. (!) Let u be a cut-vertex of a simple graph G. Prove that G - v is connected. עarrow_forward1.2.12. (-) Convert the proof at 1.2.32 to an procedure for finding an Eulerian circuit in a connected even graph.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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