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linear algebra

Linear Algebra 1st Edition Jim Hefferon - Solutions

For the matrix A below, find a set of vectors S so that the span of S equals the null space of A, (S) = N(A).
Consider the set of all size 2 vectors in the Cartesian plane R2. 1. Give a geometric description of the span of a single vector. 2. How can you tell if two vectors span the entire plane, without doing any row reduction or calculation?
Consider the set of all size 3 vectors in Cartesian 3-space R3. 1. Give a geometric description of the span of a single vector. 2. Describe the possibilities for the span of two vectors. 3. Describe the possibilities for the span of three vectors.
Letand 1. Find a vector w1, different from u and v, so that ({u, v, w1}) = ({u, v}). 2. Find a vector w2 so that ({u, v, w2}) ‰ ({u, v}).
In the spirit of Example SCAD, begin with the four columns of the coefficient matrix of Archetype C, and use these columns in a span construction to build the set S. Argue that S can be expressed as the span of just three of the columns of the coefficient matrix (saying exactly which three) and in
Suppose that v1, v2 ∈ Cm. Prove that ({v1, v2}) = ({v1, v2, 5v1 + 3v2})
Suppose that S is a set of vectors from Cm. Prove that the zero vector, 0, is an element of (S).
Solve the given vector equation for Î±, or explain why no solution exists:
Determine if the sets of vectors are linearly independent or linearly dependent. When the set is linearly dependent, exhibit a nontrivial relation of linear dependence.1.2.
For the matrix B below, find a set S that is linearly independent and spans the null space of B, that is, N(B) = (S).
For the matrix A below, find a linearly independent set S so that the null space of A is spanned by S, that is, N(A) = (S).
Find a set of column vectors, T, such that (1) the span of T is the null space of B, (T) = N(B) and (2) T is a linearly independent set.
Find a set S so that S is linearly independent and N(A) = (S), where N(A) is the null space of the matrix A below.
For the matrix A below, find a set of vectors S so that (1) S is linearly independent, and (2) the span of S equals the null space of A, (S) = N(A).
Find Î± and Î² that solve the vector equation.
Suppose that S = {v1, v2, v3} is a set of three vectors from C873. Prove that the set T = {2v1 + 3v2 + v3, v1 - v2 - 2v3, 2v1 + v2 - v3} is linearly dependent.
Suppose that S = {v1, v2, v3} is a linearly independent set of three vectors from C873. Prove that the set T = {2v1 + 3v2 + v3, v1 - v2 + 2v3, 2v1 + v2 - v3} is linearly independent.
Consider the set of vectors from C3, W, given below. Find a set T that contains three vectors from W and such that W = (T).
Consider the subspace W = ({v1, v2, v3, v4}). Find a set S so that (1) S is a subset of W, (2) S is linearly independent, and (3) W = (S). Write each vector not included in S as a linear combination of the vectors that are in S.
Suppose that {v1, v2, v3, ... , vn} is a set of vectors. Prove that {v1 - v2, v2 - v3, v3 - v4, ... , vn - v1} is a linearly dependent set.
Suppose that {v1, v2, v3, v4} is a linearly independent set in C35. Prove that {v1, v1 + v2, v1 + v2 + v3, v1 + v2 + v3 + v4} is a linearly independent set.
Suppose that A is an m × n matrix with linearly independent columns and the linear system LS(A, b) is consistent. Show that this system has a unique solution.
Find Î± and Î² that solve the vector equation.
Let T be the set of columns of the matrix B below. Define W = (T). Find a set R so that (1) R has 3 vectors, (2) R is a subset of T, and (3) W = (R).
Consider the set of vectors from C3, W, given below. Find a linearly independent set T that contains three vectors from W and such that (W) = (T).
Given the set S below, find a linearly independent set T so that (T) = (S).
Let W be the span of the set of vectors S below, W = (S). Find a set T so that 1) the span of T is W, (T) = W, (2) T is a linearly independent set, and (3) T is a subset of S.
Let T be the set of vectorsFind two different subsets of T, named R and S, so that R and S each contain three vectors, and so that (R) = (T) and (S) = (T). Prove that both R and S are linearly independent.
Suppose that v1 and v2 are any two vectors from Cm. Prove the following set equality. ({v1, v2}) = ({v1 + v2, v1 - v2})
Provide reasons (mostly vector space properties) as justification for each of the seven steps of the following proof. Theorem For any vectors u, v, w ∈ Cm, if u + v = u + w, then v = w. Proof: Let u, v, w ∈ Cm, and suppose u + v = u + w. v = 0 + v = (-u + u) + v = -u + (u + v) = -u + (u + w) =
Suppose that u, v, w ∈ Cn, α, β ∈ C and u is orthogonal to both v and w. Prove that u is orthogonal to αv + βw.
Provide reasons (mostly vector space properties) as justification for each of the six steps of the following proof. Theorem For any vector u ∈ Cm, 0u = 0: Proof: Let u ∈ Cm. 0 = 0u + (-0u) = (0 + 0)u + (-0u) = (0u + 0u) + (-0u) = 0u + (0u + (-0u)) = 0u + 0 = 0u
Provide reasons (mostly vector space properties) as justification for each of the six steps of the following proof. Theorem For any scalar c, c0 = 0. Proof: Let c be an arbitrary scalar. 0 = c0 + (-c0) = c(0 + 0) + (-c0) = (c0 + c0) + (-c0) = c0 + (c0 + (-c0)) = c0 + 0 = c0
LetPerform the following calculations: (1) A + B, (2) A + C, (3) Bt + C, (4) A + Bt, (5) Î²C, (6) 4A - 3B, (7) At + Î±C, (8) A + B - Ct, (9) 4A + 2B - 5Ct
Suppose Y is the set of all 3 × 3 symmetric matrices (Definition SYM). Find a set T so that T is linearly independent and (T) = Y.
Prove Property CM of Theorem VSPM. Write your proof in the style of the proof of Property DSAM given in this section.
Solve the given vector equation for x, or explain why no solution exists:
Compute the product of the two matrices below, AB. Do this using the definitions of the matrix-vector product (Definition MVP) and the definition of matrix multiplication (Definition MM).
Solve the given matrix equation for Î±, or explain why no solution exists:
For the matrixfind A2, A3, A4. Find a general formula for An for any positive integer n.
For the matrixfind A2, A3, A4. Find a general formula for An for any positive integer n.
For the matrixfind A2, A3, A4. Find a general formula for An for any positive integer n.
For the matrixfind A2, A3, A4. Find a general formula for An for any positive integer n.
Suppose that A is a square matrix and there is a vector, b, such that LS(A, b) has a unique solution. Prove that A is nonsingular. Give a direct proof (perhaps appealing to Theorem PSPHS) rather than just negating a sentence from the text discussing a similar situation.
Prove the second part of Theorem MMSMM.
Solve the given matrix equation for Î±, or explain why no solution exists:
Suppose that A is an m n matrix and B is an n × p matrix. Prove that the null space of B is a subset of the null space of AB, that is N(B) ⊆ N(AB). Provide an example where the opposite is false, in other words give an example where N(AB) ⊈ N(B).
Suppose that A is an n × n nonsingular matrix and B is an n × p matrix. Prove that the null space of B is equal to the null space of AB, that is N(B) = N(AB).
Give a new proof of Theorem PSPHS replacing applications of Theorem SLSLC with matrix-vector products (Theorem SLEMM).
Suppose that x, y ∈ Cn, b ∈ Cm and A is an m × n matrix. If x, y and x + y are each a solution to the linear system LS(A, b), what interesting can you say about b? Form an implication with the existence of the three solutions as the hypothesis and an interesting statement about LS(A, b) as the
If it exists, find the inverse ofand check your answer.
Find Î± and Î² that solve the following equation:
Recycle the matrices A and B from Exercise MISLE.C21 and setEmploy the matrix B to solve the two linear systems LS(A, c) and LS(A, d).
LetCompute the inverse of D, D-1, by forming the 5 Ã— 10 matrix [D | I5] and row-reducing (Theorem CINM). Then use a calculator to compute D-1 directly.
LetCompute the inverse of E, E-1, by forming the 5 Ã— 10 matrix [E | I5] and row-reducing (Theorem CINM). Then use a calculator to compute E-1 directly.
LetCompute the inverse of C, C-1, by forming the 4 Ã— 8 matrix [C | I4] and row-reducing (Theorem CINM). Then use a calculator to compute C-1 directly.
Find all solutions to the system of equations below, making use of the matrix inverse found in Exercise MISLE.C28. x1 + x2 + 3x3 + x4 = - 4 - 2x1 - x2 - 4x3 - x4 = 4 x1 + 4x2 + 10x3 + 2x4 = - 20 - 2x1 - 4x3 + 5x4 = 9
Use the inverse of a matrix to find all the solutions to the following system of equations. x1 + 2x2 - x3 = - 3 2x1 + 5x2 - x3 = - 4 - x1 - 4x2 = 2
Use a matrix inverse to solve the linear system of equations. x1 - x2 + 2x3 = 5 x1 - 2x3 = - 8 2x1 - x2 - x3 = - 6
Construct an example to demonstrate that (A + B)-1 = A-1 + B-1 is not true for all square matrices A and B of the same size.
Solve the system of equations below using the inverse of a matrix. x1 + x2 + 3x3 + x4 = 5 - 2x1 - x2 - 4x3 - x4 = - 7 x1 + 4x2 + 10x3 + 2x4 = 9 - 2x1 - 4x3 + 5x4 = 9
Find values of x, y, z so that matrixis invertible.
Find values of x, y z so that matrixis singular.
If A and B are n × n matrices, A is nonsingular, and B is singular, show directly that AB is singular, without using Theorem NPNT.
Construct an example of a 4 × 4 unitary matrix.
Show that the set S is linearly independent in M2, 2.
Matrix multiplication interacts nicely with many operations. But not always with transforming a matrix to reduced row-echelon form. Suppose that A is an m × n matrix and B is an n × p matrix. Let P be a matrix that is row-equivalent to A and in reduced row-echelon form, Q be a matrix that is
Example CSOCD expresses the column space of the coefficient matrix from Arche-type D (call the matrix A here) as the span of the first two columns of A. In Example CSMCS we determined that the vectorwas not in the column space of A and that the vector was in the column space of A. Attempt to write
For the matrix A below find a set of vectors T meeting the following requirements: (1) the span of T is the column space of A, that is, (T) = C(A), (2) T is linearly independent, and (3) the elements of T are columns of A.
Find a linearly independent set S so that the span of S, (S), is row space of the matrix B, and S is linearly independent.
Determine if the set S below is linearly independent in M2, 3.
For the 3 Ã— 4 matrix A and the column vector y ˆˆ C4 given below, determine if y is in the row space of A. In other words, answer the question: y ˆˆ R(A)?
For the matrix A below, find two different linearly independent sets whose spans equal the column space of A, C(A), such that1. The elements are each columns of A.2. The set is obtained by a procedure that is substantially different from the procedure you use in part (1).
For the matrix E below, find vectors b and c so that the system LS(E, b) is consistent and LS(E, c) is inconsistent.
Usually the column space and null space of a matrix contain vectors of different sizes. For a square matrix, though, the vectors in these two sets are the same size. Usually the two sets will be different. Construct an example of a square matrix where the column space and null space are equal.
We have a variety of theorems about how to create column spaces and row spaces and they frequently involve row-reducing a matrix. Here is a procedure that some try to use to get a column space. Begin with an m n matrix A and row-reduce to a matrix B with columns B1, B2, B3, . . . , Bn. Then form
Determine if the matrix A is in the span of S. In other words, is A ˆˆ (S)? If so write A as a linear combination of the elements of S.
Suppose that A is an m × n matrix and B is an n × p matrix. Prove that the column space of AB is a subset of the column space of A, that is C(AB) ⊆ C(A). Provide an example where the opposite is false, in other words give an example where C(A) ⊈ C(AB).
Suppose that A is an m × n matrix and B is an n × n nonsingular matrix. Prove that the column space of A is equal to the column space of AB, that is C(A) = C(AB).
Suppose that A is an m × n matrix and B is an n × m matrix where AB is a nonsingular matrix. Prove that 1. N(B) = {0} 2. C(B) ∩ N(A) = {0} Discuss the case when m = n in connection with Theorem NPNT.
Given the matrix A below, use the extended echelon form of A to answer each part of this problem. In each part, find a linearly independent set of vectors, S, so that the span of S, (S), equals the specified set of vectors1. The row space of A, R(A).2. The column space of A, C(A).3. The null space
For the matrix D below use the extended echelon form to find:1. A linearly independent set whose span is the column space of D.2. A linearly independent set whose span is the left null space of D.
For the matrix B below, find sets of vectors whose span equals the column space of B (C(B)) and which individually meet the following extra requirements.1. The set illustrates the definition of the column space.2. The set is linearly independent and the members of the set are columns of B.3. The
Let A be the matrix below, and find the indicated sets with the requested properties.1. A linearly independent set S so that C(A) = (S) and S is composed of columns of A. 2. A linearly independent set S so that C(A) = (S) and the vectors in S have a nice pattern of zeros and ones at the top of the
Suppose that V is a vector space, and u, v, w ∈ V. If w + u = w + v, then u = v.
Suppose V is a vector space, u, v ∈ V and α is a nonzero scalar from C. If αu = αv, then u = v.
Suppose V is a vector space, u ≠ 0 is a vector in V and α, β ∈ C. If αu = βu, then α = β.
Suppose that V is a vector space and α ∈ C is a scalar such that αx = x for every x ∈ V. Prove that α = 1. In other words, Property O is not duplicated for any other scalar but the "special" scalar, 1.
Working within the vector space C3, determine ifis in the subspace W,
Working within the vector space C4, determine ifis in the subspace W,
Working within the vector space C4, determine ifis in the subspace W,
Working within the vector space P3 of polynomials of degree 3 or less, determine if p(x) = x3 + 6x + 4 is in the subspace W below. W = ({x3 + x2 + x, x3 + 2x - 6, x2 - 5})
Consider the subspaceof the vector space of 2 Ã— 2 matrices, M22. Is an element of W?
Let V be the set C2 with the usual vector addition, but with scalar multiplication defined byDetermine whether or not V is a vector space with these operations.
In C3, the vector space of column vectors of size 3, prove that the set Z is a subspace.
A square matrix A of size n is upper triangular if [A]ij = 0 whenever i > j. Let UTn be the set of all upper triangular matrices of size n. Prove that UTn is a subspace of the vector space of all square matrices of size n, Mnn.
Let P be the set of all polynomials, of any degree. The set P is a vector space. Let E be the subset of P consisting of all polynomials with only terms of even degree. Prove or disprove: the set E is a subspace of P.
Let P be the set of all polynomials, of any degree. The set P is a vector space. Let F be the subset of P consisting of all polynomials with only terms of odd degree. Prove or disprove: the set F is a subspace of P.
In the vector space of 2 Ã— 2 matrices, M22, determine if the set S below is linearly independent.
In the crazy vector space C (Example CVS), is the set S = {(0, 2), (2, 8)} linearly independent?

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