Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))
9th Edition
ISBN: 9780321962218
Author: Steven J. Leon
Publisher: PEARSON
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Chapter 4, Problem 3CTB
a.
To determine
To find:The kernel of
b.
To determine
To find:The
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3. Let F be a subfield of complex numbers. Let V be the vector space (over F), consisting
of all polynomials, i.e. V = F[r]. Let T:V → V be the function defined by
T: f(x) + (x + 1)f'(x).
d
For example T(x² + x) = (x + 1) . — (x² + x) = (x + 1)(2x + 1) = 2x² + 3x +1.
dr
(a) Is T a linear transformation? Give proper reasons.
(b) Is T invertible? Give proper reasons.
For each linear operator T on V, find the minimal polynomial of T.
(a) V = R2 and T(a, b) = (a + b, a −b)
(b) V = P2(R) and T(g(x)) = g’(x) + 2g(x)
(c) V = P2(R) and T(f(x)) = −xf”(x) + f’(x) + 2f(x)
(d) V = Mn×n(R) and T(A) = At. Hint: Note that T2 = I.
Provide an example of two distinct linear operators T1 and T2 on an inner product space V such that (given question)
Justify your answer.
Chapter 4 Solutions
Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))
Ch. 4.1 - Show that each of the following are linear...Ch. 4.1 - Let L be the linear operator on 2 defined by...Ch. 4.1 - Let a be a fixed nonzero vector in 2 . A mapping...Ch. 4.1 - Let L: 22 be a linear operator. If...Ch. 4.1 - Determine whether the following are linear...Ch. 4.1 - Determine whether the following are linear...Ch. 4.1 - Determine whether the following are linear...Ch. 4.1 - Let C be a fixed nn matrix. Determine whether the...Ch. 4.1 - Determine whether the following are linear...Ch. 4.1 - For each fC[0,1] , define L(f)=F , where F(x)= 0...
Ch. 4.1 - Determine whether the following are linear...Ch. 4.1 - Use mathematical induction to prove that if L is a...Ch. 4.1 - Let {v1,...,vn} be a basis for a vector space V,...Ch. 4.1 - Let L be a linear operator on 1 and let a=L(1) ....Ch. 4.1 - Let L be a linear operator on a vector space V....Ch. 4.1 - Let L1:UV and L2:VW be a linear transformations,...Ch. 4.1 - Determine the kernel and range of each of the...Ch. 4.1 - Let S be the subspace of 3 spanned by e1 and e2 ....Ch. 4.1 - Find the kernel and range of each of the following...Ch. 4.1 - Let L:VW be a linear transformation, and let T be...Ch. 4.1 - A linear transformation L:VW is said to be...Ch. 4.1 - A linear transformation L:VW is said to be map V...Ch. 4.1 - Which of the operators defined in Exercise 17 are...Ch. 4.1 - Let A be a 22 matrix, and let LA be the linear...Ch. 4.1 - Let D be the differentiation operator on P3 , and...Ch. 4.2 - Refer to Exercise 1 of Section 4.1. For each...Ch. 4.2 - For each of the following linear transformations L...Ch. 4.2 - For each of the following linear operators L on 3...Ch. 4.2 - Let L be the linear operators on 3 defined by...Ch. 4.2 - Find the standard matrix representation for each...Ch. 4.2 - Let b1=[110],b2=[101],b3=[011] and let L be the...Ch. 4.2 - Let y1=[111],y2=[110],y3=[100] and let I be the...Ch. 4.2 - Let y1,y2, and y3 be defined as in Exercise 7, and...Ch. 4.2 - Let R=[001100110011111] The column vectors of R...Ch. 4.2 - For each of the following linear operators on 2 ,...Ch. 4.2 - Determine the matrix representation of each of the...Ch. 4.2 - Let Y, P, and R be the yaw, pitch, and roll...Ch. 4.2 - Let L be the linear transformatino mapping P2 into...Ch. 4.2 - The linear transformation L defined by...Ch. 4.2 - Let S be the subspace of C[a,b] spanned by ex,xex...Ch. 4.2 - Let L be the linear operator on n . Suppose that...Ch. 4.2 - Let L be a linear operator on a vector space V....Ch. 4.2 - Let E=u1,u2,u3 and F=b1,b2 , where...Ch. 4.2 - Suppose that L1:VW and L2:WZ are linear...Ch. 4.2 - Let V and W be vector spaces with ordered bases E...Ch. 4.3 - For each of the following linear operators L on 2...Ch. 4.3 - Let u1,u2 and v1,v2 be ordered bases for 2 , where...Ch. 4.3 - Let L be the linear transformation on 3 defined by...Ch. 4.3 - Let L be the linear operator mapping 3 into 3...Ch. 4.3 - Let L be the operator on P3 defined by...Ch. 4.3 - Let V be the subspace of C[a,b] spanned by 1,ex,ex...Ch. 4.3 - Prove that if A is similar to B and B is similar...Ch. 4.3 - Suppose that A=SS1 , where is a diagonal matrix...Ch. 4.3 - Suppose that A=ST , where S is nonsingular. Let...Ch. 4.3 - Let A and B be nn matrices. Show that is A is...Ch. 4.3 - Show that if A and B are similar matrices, then...Ch. 4.3 - Let A and B t similar matrices. Show that (a) AT...Ch. 4.3 - Show that if A is similar to B and A is...Ch. 4.3 - Let A and B be similar matrices and let be any...Ch. 4.3 - The trace of an nn matrix A, denoted tr(A) , is...Ch. 4 - Use MATLAB to generate a matrix W and a vector x...Ch. 4 - Set A=triu(ones(5))*tril(ones(5)) . If L denotes...Ch. 4 - Prob. 3ECh. 4 - For each statement that follows, answer true if...Ch. 4 - Prob. 2CTACh. 4 - Prob. 3CTACh. 4 - For each statement that follows, answer true if...Ch. 4 - Prob. 5CTACh. 4 - Prob. 6CTACh. 4 - Prob. 7CTACh. 4 - Prob. 8CTACh. 4 - Prob. 9CTACh. 4 - Prob. 10CTACh. 4 - Determine whether the following are linear...Ch. 4 - Prob. 2CTBCh. 4 - Prob. 3CTBCh. 4 - Prob. 4CTBCh. 4 - Prob. 5CTBCh. 4 - Prob. 6CTBCh. 4 - Let L be the translation operator on 2 defined by...Ch. 4 - Let u1=[ 3 1 ],u2=[ 5 2 ] and let L be the linear...Ch. 4 - Let
and
and let L be the linear operator onwhose...Ch. 4 - Prob. 10CTB
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- Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are linearly dependent in the vector space C[0,1], but linearly independent in C[1,1].arrow_forwardLet T be a linear transformation from P2 into P2 such that T(1)=x,T(x)=1+xandT(x2)=1+x+x2. Find T(26x+x2).arrow_forwardLet T be a linear transformation T such that T(v)=kv for v in Rn. Find the standard matrix for T.arrow_forward
- Let T:R4R2 be the linear transformation defined by T(v)=Av, where A=[10100101]. Find a basis for a the kernel of T and b the range of T. c Determine the rank and nullity of T.arrow_forwardLet T:R3R3 be the linear transformation that projects u onto v=(2,1,1). (a) Find the rank and nullity of T. (b) Find a basis for the kernel of T.arrow_forwardLet A and B be square matrices of order n satisfying, Ax=Bx for all x in all Rn. a Find the rank and nullity of AB. b Show that matrices A and B must be identical.arrow_forward
- Let f be the function from {a, b, c} to {1, 2, 3} such that f (a) = 2, f (b) = 3, and f (c) = 1. Is f invertible, and if it is, what is its inverse?arrow_forwardLet P3 be the vector space of all polynomials of degree 3 or less in the variable z. Let = 2+x+x², 2+x+x², 2+x², = 11 + 3x + 6x² choose PI(T) P2(x) P3(x) = P4(x) and let C = {p1(x), p2(x), P3(x), P4(x)}. a. Use coordinate representations with respect to the basis B = {1, 2, ², ³} to determine whether the set C forms a basis for P.. = c. The dimension of span(C) is b. Find a basis for span(C). Enter a polynomial or a comma separated list of polynomials. {}arrow_forwardLet I be the linear operator in R2 defined by Find bases of the kernel and image of L. Kernel: L(x) = (6x1 — 9x2, 2x1 – 3x₂)T Image:arrow_forward
- Let 1 A = [23] 4 Recall that row (A) and nul(A) are orthogonal complements. So we can split any x E R² into a unique component x, in row(A) and a unique component Xn in nul(A) such that X = Xr+ Xn.arrow_forward. Let V be the set of all pairs (x,y) of real numbers together with the following operations: (x1,y1) (x2,y2) = (4x1 + x2 − 4, Y1 + 3 y2 − 3) c(x,y) = (cx- c+1, cy − c + 1). (a) Show that scalar multiplication distributes over vector addition, that is: c((x1,y1)(x2,y2)) = (c○ (x1,y1)) + (c○ (x2,y2)). (b) Explain why V nonetheless is not a vector space by showing that a vector space property does not hold for this set with these operations.arrow_forward
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