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mathematics
linear algebra
Linear Algebra 1st Edition Jim Hefferon - Solutions
In the vector space of polynomials P3, determine if the set S is linearly independent or linearly dependent. S = {2 + x - 3x2 - 8x3, 1 + x + x2 + 5x3, 3 - 4x2 - 7x3}
Determine if the set S = {(3, 1), (7, 3)} is linearly independent in the crazy vector space C (Example CVS).
In the vector space of real-valued functions F = {f | f : R → R}, determine if the following set S is linearly independent. S = {sin2 x, cos2 x, 2}
Let V be the set C2 with the usual scalar multiplication, but with vector addition defined byDetermine whether or not V is a vector space with these operations.
Let1. Determine if S spans M2,2. 2. Determine if S is linearly independent.
Let1. Determine if S spans M2,2. 2. Determine if S is linearly independent.
Determine if the set T = {x2 - x + 5, 4x3 - x2 + 5x, 3x + 2} spans the vector space of polynomials with degree 4 or less, P4.
The set W is a subspace of M22, the vector space of all 2 × 2 matrices. Prove that S is a spanning set for W.
Determine if the set S = {(3, 1), (7, 3)} spans the crazy vector space C (Example CVS).
Halfway through Example SSP4, we need to show that the system of equationsis consistent for every choice of the vector of constants satisfying 16a + 8b + 4c + 2d + e = 0. Express the column space of the coeffocient matrix of this system as a null space, using Theorem FS. From this use Theorem CSCS
Suppose that S is a finite linearly independent set of vectors from the vector space V. Let T be any subset of S. Prove that T is linearly independent.
Suppose that V is a vector space and u, v ∈ V are two vectors in V. Use the definition of linear independence to prove that S = {u, v} is a linearly dependent set if and only if one of the two vectors is a scalar multiple of the other. Prove this directly in the context of an abstract vector
Carefully formulate the converse of Theorem VRRB and provide a proof.
Let V be the set M2,2 with the usual scalar multiplication, but with addition defined by A + B = O2,2 for all 2 × 2 matrices A and B. Determine whether or not V is a vector space with these operations.
Find a basis for (S), where
Find a basis for the subspace W of C4,
Find a basis for the vector space T of lower triangular 3 × 3 matrices; that is, matrices of the formwhere an asterisk represents any complex number.
Find a basis for the subspace Q of P2, Q = {p(x) = a + bx + cx2 | p(0) = 0}.
Find a basis for the subspace R of P2, R = {p(x) = a + bx + cx2 | p′(0) = 0}, where p′ denotes the derivative.
From Example RSB, form an arbitrary (and nontrivial) linear combination of the four vectors in the original spanning set for W. So the result of this computation is of course an element of W. As such, this vector should be a linear combination of the basis vectors in B. Find the (unique) scalars
In Example BM provide the verifications (linear independence and spanning) to show that B is a basis of Mmn.
Theorem UMCOB says that unitary matrices are characterized as those matrices that "carry" orthonormal bases to orthonormal bases. This problem asks you to prove a similar result: nonsingular matrices are characterized as those matrices that "carry" bases to bases. More precisely, suppose that A is
Let V be the set M2,2 with the usual addition, but with scalar multiplication defined by αA = O2,2 for all 2 × 2 matrices A and scalars α. Determine whether or not V is a vector space with these operations.
Find the dimension of the subspaceof C4.
Find the dimension of the subspace W = {a + bx + cx2 + dx3 | a + b + c + d = 0} of P3.
Find the dimension of the subspaceof M2,2.
For the matrix A below, compute the dimension of the null space of A, dim (N(A)).
The set W below is a subspace of C4. Find the dimension of W.
Find the rank and nullity of the matrix
Find the rank and nullity of the matrix
Consider the following sets of 3×3 matrices, where the symbol * indicates the position of an arbitrary complex number. Determine whether or not these sets form vector spaces with the usual operations of addition and scalar multiplication for matrices.1. All matrices of the form2. All
M22 is the vector space of 2 × 2 matrices. Let S22 denote the set of all 2 × 2 symmetric matrices. That is S22 = {A ∈ M22 | At = A} 1. Show that S22 is a subspace of M22. 2. Exhibit a basis for S22 and prove that it has the required properties. 3. What is the dimension of S22?
A 2 × 2 matrix B is upper triangular if [B]21 = 0. Let UT2 be the set of all 2 × 2 upper triangular matrices. Then UT2 is a subspace of the vector space of all 2 × 2 matrices, M22 (you may assume this). Determine the dimension of UT2 providing all of the necessary justifications for your answer.
Suppose that A is an m × n matrix and b ∈ Cm. Prove that the linear system LS(A, b) is consistent if and only if r (A) = r ([A | b]).
Part of Exercise B.T50 is the half of the proof where we assume the matrix A is nonsingular and prove that a set is basis. In Solution B.T50 we proved directly that the set was both linearly independent and a spanning set. Shorten this part of the proof by applying Theorem G. Be careful, there is
Suppose that W is a vector space with dimension 5, and U and V are subspaces of W, each of dimension 3. Prove that U ∩ V contains a non-zero vector. State a more general result.
The set of integers is denoted Z. Does the setwith the operations of standard addition and scalar multiplication of vectors form a vector space?
Doing the computations by hand, find the determinant of the matrix below.
Doing the computations by hand, find the determinant of the matrix A.
Find a value of k so that the matrixhas det(A) = 0, or explain why it is not possible.
Find a value of k so that the matrixhas det(A) = 0, or explain why it is not possible.
Given the matrixfind all values of x that are solutions of det(B) = 0
Given the matrixfind all values of x that are solutions of det(B) = 0.
Construct a 3 × 3 nonsingular matrix and call it A. Then, for each entry of the matrix, compute the corresponding cofactor, and create a new 3 × 3 matrix full of these cofactors by placing the cofactor of an entry in the same location as the entry it was based on. Once complete, call this matrix
Doing the computations by hand, find the determinant of the matrix below.
Doing the computations by hand, find the determinant of the matrix below.
Doing the computations by hand, find the determinant of the matrix below.
Doing the computations by hand, find the determinant of the matrix below.
Doing the computations by hand, find the determinant of the matrix A.
Doing the computations by hand, find the determinant of the matrix A.
Doing the computations by hand, find the determinant of the matrix A.
Doing the computations by hand, find the determinant of the matrix A.
Find the eigenvalues, eigenspaces, algebraic and geometric multiplicities for the 3 × 3 identity matrix I3. Do your results make sense?
For matrixthe characteristic polynomial of A is pA (λ) = (4 - x)(1 - x)2. Find the eigenvalues and corresponding eigenspaces of A.
For matrixthe characteristic polynomial of A is pA(λ) = (x + 2)(x - 2)2(x - 4): Find the eigenvalues and corresponding eigenspaces of A.
Repeat Example CAEHW by choosingand then arrive at an eigenvalue and eigenvector of the matrix A. The hard way.
A matrix A is idempotent if A2 = A. Show that the only possible eigenvalues of an idempotent matrix are λ = 0 and λ = 1. Then give an example of a matrix that is idempotent and has both of these two values as eigenvalues.
The characteristic polynomial of the square matrix A is usually defined as rA(x) = det (xIn - A). Find a specific relationship between our characteristic polynomial, pA (x), and rA(x), give a proof of your relationship, and use this to explain why Theorem EMRCP can remain essentially unchanged with
Suppose that λ and ρ are two different eigenvalues of the square matrix A. Prove that the intersection of the eigenspaces for these two eigenvalues is trivial. That is, εA (λ) ∩ εA (ρ) = {0}.
Suppose that A is a square matrix. Prove that the constant term of the characteristic polynomial of A is equal to the determinant of A.
Suppose that A is a square matrix. Prove that a single vector may not be an eigenvector of A for two different eigenvalues.
Theorem EIM says that if λ is an eigenvalue of the nonsingular matrix A, then 1 λ is an eigenvalue of A-1. Write an alternate proof of this theorem using the characteristic polynomial and without making reference to an eigenvector of A for λ.
Consider the matrix A below. First, show that A is diagonalizable by computing the geometric multiplicities of the eigenvalues and quoting the relevant theorem. Second, find a diagonal matrix D and a nonsingular matrix S so that S-1AS = D.
Determine if the matrix A below is diagonalizable. If the matrix is diagonalizable, then find a diagonal matrix D that is similar to A, and provide the invertible matrix S that performs the similarity transformation. You should use your calculator to find the eigenvalues of the matrix, but try only
Consider the matrix A below. Find the eigenvalues of A using a calculator and use these to construct the characteristic polynomial of A, pA (x). State the algebraic multiplicity of each eigenvalue. Find all of the eigenspaces for A by computing expressions for null spaces, only using your
Suppose that A and B are similar matrices. Prove that A3 and B3 are similar matrices. Generalize.
Suppose that A and B are similar matrices, with A nonsingular. Prove that B is nonsingular, and that A-1 is similar to B-1.
Suppose that B is a nonsingular matrix. Prove that AB is similar to BA.
Find the eigenvalues, eigenspaces, algebraic multiplicities and geometric multiplicities for the matrix below. It is possible to do all these computations by hand, and it would be instructive to do so.
Find the eigenvalues, eigenspaces, algebraic multiplicities and geometric multiplicities for the matrix below. It is possible to do all these computations by hand, and it would be instructive to do so.
The matrix A below has λ = 2 as an eigenvalue. Find the geometric multiplicity of λ = 2 using your calculator only for row-reducing matrices.
Without using a calculator, find the eigenvalues of the matrix B.
Find the eigenvalues, eigenspaces, algebraic and geometric multiplicities for
Find the eigenvalues, eigenspaces, algebraic and geometric multiplicities for A =
Define T: M2,2 †’ R byFind the pre-image T-1 (3).
Define T: P3 → P2 by T (a + bx + cx2 + dx3) = b + 2cx + 3dx2. Find the pre-image of 0. Does this linear transformation seem familiar?
Equivalence relations always create a partition of the set they are defined on, via a construction called equivalence classes. For the relation in the previous problem, the equivalence classes are the pre-images. Prove directly that the collection of pre-images partition U by showing that (a) every
The linear transformation T: C4 †’ C3 is not injective. Find two inputs x, y ˆˆ C4 that yield the same output (that is T (x) = T (y)).
Define the linear transformationFind a basis for the kernel of T, K(T). Is T injective?
Letand let T: C5 †’ C4 be given by T (x) = Ax. Is T injective?
Let T: C3 †’ C3 be given byFind K(T). Is T injective?
Letand let T: C4 †’ C4 be given by T (x) = Ax. Find K(T). Is T injective?
Letand let T: C4 †’ C4 be given by T (x) = Ax. Find K(T). Is T injective?
Let T: M2,2 †’ P2 be given byIs T injective? Find K(T).
Given that the linear transformation T: C3 †’ C3,is injective, show directly that {T (e1), T (e2), T (e3)} is a linearly independent set.
Given that the linear transformation T: C2 †’ C3,is injective, show directly that {T (e1), T (e2)} is a linearly independent set.
Given that the linear transformation T: C3 †’ C5,is injective, show directly that {T (e1), T (e2), T (e3)} is a linearly independent set.
Show that the linear transformation R is not injective by finding two different elements of the domain, x and y, such that R(x) = R(y). (S22 is the vector space of symmetric 2 × 2 matrices.)
Suppose that that T: U → V and S: V → W are linear transformations. Prove the following relationship between kernels. K(T) ⊆ K(S ο T)
The linear transformation S: C4 †’ C3 is not surjective. Find an output w ˆˆ C3 that has an empty pre-image (that is S-1 (w) = ˆ….)
Determine whether or not the following linear transformation T: C5 †’ P3 is surjective:
Determine whether or not the linear transformation T: P3 †’ C5 below is surjective:
Define the linear transformationFind a basis for the range of T, R(T). Is T surjective?
Let T: C3 †’ C3 be given byFind a basis of R(T). Is T surjective?
Let T: C3 †’ C4 be given byFind a basis of R(T). Is T surjective?
Define the linear transformationVerify that T is a linear transformation.
Let T: C4 †’ M2,2 be given byFind a basis of R(T). Is T surjective?
Let T: P2 → P4 be given by T (p(x)) = x2p(x). Find a basis of R(T). Is T surjective?
Let T: P4 → P3 be given by T (p(x)) = p′(x), where p′(x) is the derivative. Find a basis of R(T). Is T surjective?
Show that the linear transformation T is not surjective by finding an element of the codomain, v, such that there is no vector u with T (u) = v.
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