Linear Algebra Exam 2
Let H be the set of all points of the form (s,s−1). Determine whether H is a vector space. If it is not a vector space, determine which of the following properties it fails to satisfy. 1. Contains a zero vector 2. Closed under vector addition 3. Closed under multiplication by scalars
H is not a vector space because fails to satisfy all three properties.
Let H be the set of all polynomials of the form p(t)=a+bt2 where a and b are in ℝ and b>a. Determine whether H is a vector space. If it is not a vector space, determine which of the following properties it fails to satisfy. 1. Contains a zero vector 2. Closed under vector addition 3. Closed under multiplication by scalars
H is not a vector space because it is not closed under multiplication by scalars and does not contain a zero vector.
Show that if A is both diagonalizable and invertible, then so is A−1. What does it mean if A is diagonalizable? What does it mean if A is invertible? What is the inverse of A? Therefore, A−1 is also diagonalizable.
If A is diagonalizable, then A=PDP^−1 for some invertible P and diagonal D. Zero is not an eigenvalue of A, so the diagonal entries in D are not zero, so D is invertible. A^−1=(PD^-1)P^1
Show that if A^2 is the zero matrix, then the only eigenvalue of A is 0.
If Ax=λx for some x≠0, then 0x=A2x=A(Ax)=A(λx)=λAx=λ2x=0. Since x is nonzero, λ must be zero. Thus, each eigenvalue of A is zero.
Let A be an n×n matrix. Determine whether the statement below is true or false. Justify the answer. A matrix A is diagonalizable if A has n eigenvectors.
The statement is false. A diagonalizable matrix must have n linearly independent eigenvectors.
A is an n×n matrix. Determine whether the statement below is true or false. Justify the answer. To find the eigenvalues of A, reduce A to echelon form.
The statement is false. An echelon form of a matrix A usually does not display the eigenvalues of A.
Determine whether the statement below is true or false. Justify the answer. B is a basis for a vector space V. The correspondence [x]B↦x is called the coordinate mapping.
The statement is false. By the definition, the correspondence x↦[x]B is called the coordinate mapping.
A is an n×n matrix. Determine whether the statement below is true or false. Justify the answer. The eigenvalues of a matrix are on its main diagonal.
The statement is false. If the matrix is a triangular matrix, the values on the main diagonal are eigenvalues. Otherwise, the main diagonal may or may not contain eigenvalues.
Let A be an n×n matrix. Determine whether the statement below is true or false. Justify the answer. If λ+5 is a factor of the characteristic polynomial of A, then 5 is an eigenvalue of A.
The statement is false. If λ+5 is a factor of the characteristic polynomial of A, then −5 is an eigenvalue of A. In order for 5 to be an eigenvalue of A, the characteristic polynomial would need to have a factor of λ−5.
Let B and C be bases for a vector space V. Determine whether the statement below is true or false. Justify the answer. If V=ℝ2, B=b1,b2, and C=c1,c2, then row reduction of c1c2b1b2 to IP produces a matrix P that satisfies [x]B=P[x]C for all x in V.
The statement is false. Matrix P satisfies [x]C=P[x]B for all x in V.
A denotes an m×n matrix. Determine whether the statement is true or false. Justify your answer: The column space of A, Col A, is the set of all solutions of Ax=b.
The statement is false. The column space of A is Col A={b : b=Ax for some x in ℝn}.
Let A be an m×n matrix. Determine whether the statement below is true or false. Justify the answer. The number of variables in the equation Ax=0 equals the nullity of A.
This statement is false. The number of free variables is equal to the nullity of A.
Let V be a nonzero finite-dimensional vector space, and the vectors listed belong to V. Determine whether the statement below is true or false. Justify the answer. If there exists a set {v1, ... , vp} that spans V, then dim V≤p.
This statement is true. Apply the Spanning Set Theorem to the set {v1, ... , vp} and produce a basis for V. This basis will not have more than p elements in it, so dim V≤p.
Suppose a 5×7 matrix A has two pivot columns. What is nullity A? Is Col A=ℝ2? Why or why not?
nullity A=5 Is Col A=ℝ2? Yes, because the number of pivot positions in A is 2.
If the null space of a 9×8 matrix is 3-dimensional, find rank A, dim Row A, and dim Col A.
rank A=5, dim Row A=5, dim Col A=5
Let V be the vector space of functions of the form y(t)=c1cosωt+c2sinωt, where ω is a fixed constant and c1 and c2 are arbitrary (varying) constants. Find a basis for V.
{cosωt, sinωt}
Suppose an 8×10 matrix A has eight pivot columns. Is Nul A=ℝ2?
No, Nul A is not equal to ℝ2. It is true that dim Nul A=2, but Nul A is a subspace of ℝ10.
Let V be the vector space of functions of the form y(t)=c1cosωt+c2sinωt, where ω is a fixed constant and c1 and c2 are arbitrary (varying) constants. Find a basis for V.
{cosωt, sinωt}
If A is a 5×7 matrix, what is the smallest possible dimension of Nul A?
2
In the vector space of all real-valued functions, find a basis for the subspace spanned by {sint,sin2t,sintcost}.
A basis for this subspace is {sint, sin2t}.
In the vector space of all real-valued functions, find a basis for the subspace spanned by {sint,sin2t,sintcost}.
A basis for this subspace is {sint,sin2t}
Which of the following sets is a subspace of ℙn for an appropriate value of n? A. All polynomials of the form p(t)=a+bt2, where a and b are in ℝ B. All polynomials of degree exactly 4, with real coefficients C. All polynomials of degree at most 4, with positive coefficients
A only
Which of the sets of vectors below are linearly independent? A. The set p1, p2, p3, where p1(t)=1, p2(t)=t2, p3(t)=5+2t. B. The set p1, p2, p3, where p1(t)=t, p2(t)=t2, p3(t)=4t+2t2. C. The set p1, p2, p3, where p1(t)=1, p2(t)=t2, p3(t)=5+2t+t2.
A. The set p1, p2, p3, where p1(t)=1, p2(t)=t2, p3(t)=5+2t. C. The set p1, p2, p3, where p1(t)=1, p2(t)=t2, p3 (t)=5+2t+t2.
Which of the sets of vectors below are linearly independent? A. The set {sint,tant} in C[0,1]. B. The set {sintcost,cos2t} in C[0,1]. C. The set cos2t, 1+cos2t in C[0,1].
A. The set {sint,tant} in C[0,1]. B. The set {sintcost,cos2t} in C[0,1].
Which of the following statements is false? A. The dimension of the vector space ℙ6 of polynomials is 7. B. Any line in ℝ3 is a one-dimensional subspace of ℝ3. C. If a vector space V has a basis B=b1,...,b5, then any set in V containing 6 vectors must be linearly dependent.
B. Any line in ℝ3 is a one-dimensional subspace of ℝ3.
Given the set of vectors S=100,012, which of the following statements are true? A. S is linearly independent and spans ℝ3. S is a basis for ℝ3. B. S is linearly independent but does not span ℝ3. S is not a basis for ℝ3. C. S spans ℝ3 but is not linearly independent. S is not a basis for ℝ3. D. S is not linearly independent and does not span ℝ3. S is not a basis for ℝ3.
B. S is linearly independent but does not span ℝ3. S is not a basis for ℝ3.
Which of the following statements is true? A. If V is a 3-dimensional vector space, then any set of exactly 3 elements in V is automatically a basis for V. B. If there exists a set v1,...,v3 that spans V, then dim V=3. C. If H is a subspace of a finite-dimensional vector space V, then dim H≤dim V.
C. If H is a subspace of a finite-dimensional vector space V, then dim H≤dim V.
Suppose ℝ4=Span{v1,...,v4}. Explain why {v1,...,v4} is a basis for ℝ4.
Let A=[v1 v2 v3 v4]. Note that A is a 4×4 matrix and its columns span ℝ4. Thus, by the Invertible Matrix Theorem, the columns are linearly independent. Therefore, the columns of A are a basis for ℝ4 because of the definition of a basis.
For this exercise, refer to the vector space of signals, S. The sift transformation, Syk=yk−1, shifts each entry in the signal one position to the right. The moving average transformation, M2yk=yk+yk−12, creates a new signal by averaging two consecutive terms in the given signal. The constant signal of all ones is given by χ=1k and the alternating signal by α={(−1)k}. Show that α is an eigenvector of the moving average transformation M2. What is the associated eigenvalue?
M2(α)=0 so α is an eigenvector of the moving average transformation M2 with eigenvalue 0
A is a 3×3 matrix with two eigenvalues. Each eigenspace is one-dimensional. Is A diagonalizable? Why?
No. The sum of the dimensions of the eigenspaces equals 2 and the matrix has 3 columns. The sum of the dimensions of the eigenspace and the number of columns must be equal.
For this exercise, refer to the vector space of signals, S. The sift transformation, Syk=yk−1, shifts each entry in the signal one position to the right. The moving average transformation, M2yk=yk+yk−12, creates a new signal by averaging two consecutive terms in the given signal. The constant signal of all ones is given by χ=1k and the alternating signal by α={(−1)k}. Show that χ is an eigenvector of the shift transformation S. What is the associated eigenvalue?
S(χ)=χ so χ is an eigenvector of the shift transformation S with eigenvalue 1
Use a property of determinants to show that A and A^T have the same characteristic polynomial.
Start with detAT−λI)=detAT−λIT)=det(A−λI)T. Then use the formula det AT=det A.
If A is a 10×8 matrix, what is the largest possible rank of A? If A is an 8×10 matrix, what is the largest possible rank of A? Explain your answers.
The rank of A is equal to the number of pivot positions in A. Since there are only 8 columns in a 10×8 matrix, and there are only 8 rows in an 8×10 matrix, there can be at most 88 pivot positions for either matrix. Therefore, the largest possible rank of either matrix is 88.
Determine if the given set is a subspace of ℙ2. Justify your answer. The set of all polynomials of the form p(t)=at2, where a is in ℝ.
The set is a subspace of ℙ2. The set contains the zero vector of ℙ2, the set is closed under vector addition, and the set is closed under multiplication by scalars.
Determine if the given set is a subspace of ℙ3. Justify your answer: The set of all polynomials of the form p(t)=at^3, where a is in ℝ.
The set is a subspace of ℙ3. The set contains the zero vector of ℙ3, the set is closed under vector addition, and the set is closed under multiplication by scalars.
Determine if the given set is a subspace of ℙn. Justify your answer. The set of all polynomials in ℙn such that p(0)=0
The set is a subspace of ℙn because the set contains the zero vector of ℙn, the set is closed under vector addition, and the set is closed under multiplication by scalars.
Determine if the following statement is true or false. Justify the answer. If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A.
The statement is false because the columns of an echelon form B of A are not necessarily in the column space of A.
Determine if the following statement is true or false. Justify the answer. The standard method for producing a spanning set for Nul A sometimes fails to produce a basis for Nul A.
The statement is false because the method always produces an independent set.
Determine if the following statement is true or false. Justify the answer. A linearly independent set in a subspace H is a basis for H.
The statement is false because the subspace spanned by the set must also coincide with H.
Let B and C be bases for a vector space V. Determine whether the statement below is true or false. Justify the answer. The columns of the change-of-coordinates matrix PC←B are B-coordinate vectors of the vectors in C.
The statement is false. The columns of the matrix PC←B are the C-coordinate vectors of the vectors in B.
A is an n×n matrix. Determine whether the statement below is true or false. Justify the answer. If Ax=λx for some scalar λ, then x is an eigenvector of A.
The statement is false. The condition that Ax=λx for some scalar λ is not sufficient to determine if x is an eigenvector of A. The vector x must be nonzero.
A is an n×n matrix. Determine whether the statement below is true or false. Justify the answer. If Ax=λx for some vector x, then λ is an eigenvalue of A.
The statement is false. The condition that Ax=λx for some vector x is not sufficient to determine if λ is an eigenvalue. The equation Ax=λx must have a nontrivial solution.
Let A, P, and D be n×n matrices. Determine whether the statement below is true or false. Justify the answer. A is diagonalizable if and only if A has n eigenvalues, counting multiplicities.
The statement is false. The eigenvalues of A may not produce enough eigenvectors to form a basis of
Let A, P, and D be n×n matrices. Determine whether the statement below is true or false. Justify the answer. A is diagonalizable if A=PDP−1 for some matrix D and some invertible matrix P.
The statement is false. The symbol D does not automatically denote a diagonal matrix.
A is an n×n matrix. Determine whether the statement below is true or false. Justify the answer. If v1 and v2 are linearly independent eigenvectors, then they correspond to distinct eigenvalues.
The statement is false. There may be linearly independent eigenvectors that both correspond to the same eigenvalue.
A denotes an m×n matrix. Determine whether the statement is true or false. Justify your answer. The row space of AT is the same as the column space of A.
The statement is true because the rows of A^T are the columns of (A^T)^T=A.
Determine if the following statement is true or false. Justify the answer. A basis is a linearly independent set that is as large as possible.
The statement is true by the definition of a basis.
Determine whether the statement below is true or false. Justify the answer. In some cases, a plane in ℝ3 can be isomorphic to ℝ2.
The statement is true. A plane in ℝ3 that passes through the origin is isomorphic to ℝ2.
Determine whether the statement is True or False. Justify your answer: If u is a vector in a vector space V, then (−1)u is the same as the negative of u.
The statement is true. For each u in V, there is a vector −u in V such that −u=(−1)u.
A is an n×n matrix. Determine whether the statement below is true or false. Justify the answer. A matrix A is not invertible if and only if 0 is an eigenvalue of A.
The statement is true. If 0 is an eigenvalue of A, then there are nontrivial solutions to the equation Ax=0x. The equation Ax=0x is equivalent to the equation Ax=0, and Ax=0 has nontrivial solutions if and only if A is not invertible.
Determine if the following statement is true or false. Justify the answer. If A and B are row equivalent, then their row spaces are the same.
The statement is true. If B is obtained from A by row operations, the rows of B are linear combinations of the rows of A and vice-versa.
Let B and C be bases for a vector space V. Determine whether the statement below is true or false. Justify the answer. If V=ℝn and C is the standard basis for V, then PC←B is the same as the change-of-coordinates matrix PB that satisfies x=PB[x]B for all x in V.
The statement is true. If C is the standard basis for ℝn, then biC=bi for 1≤i≤n, and PB=b1...bn.
Let A be an n×n matrix. Determine whether the statement below is true or false. Justify the answer. The multiplicity of a root r of the characteristic equation of A is called the algebraic multiplicity of r as an eigenvalue of A.
The statement is true. It is the definition of the algebraic multiplicity of an eigenvalue of A.
Assume A, P, and D are n×n matrices. Determine whether the statement below is true or false. Justify the answer. If AP=PD, with D diagonal, then the nonzero columns of P must be eigenvectors of A.
The statement is true. Let v be a nonzero column in P and let λ be the corresponding diagonal element in D. Then AP=PD implies that Av=λv, which means that v is an eigenvector of A.
Determine whether the statement is True or False. Justify your answer: A vector space is also a subspace of itself.
The statement is true. The axioms for a vector space include all the conditions for being a subspace.
Let B and C be bases for a vector space V. Determine whether the statement below is true or false. Justify the answer. The columns of PC←B are linearly independent.
The statement is true. The columns of PC←B are linearly independent because they are the coordinate vectors of the linearly independent set B.
Determine whether the statement is True or False. Justify your answer: A vector is any element of a vector space.
The statement is true. The elements of a vector space are called vectors.
A denotes an m×n matrix. Determine whether the statement is true or false. Justify your answer. A null space is a vector space.
The statement is true. The null space of an m×n matrix A is a subspace of ℝn.
A denotes an m×n matrix. Determine whether the statement is true or false. Justify your answer: A null space is a vector space.
The statement is true. The null space of an m×n matrix A is a subspace of ℝn.
Let A be an m×n matrix. Determine whether the statement below is true or false. Justify the answer. The nullity of A is the number of columns of A that are not pivot columns.
The statement is true. The nullity of A equals the number of free variables in the equation Ax=0.
Determine whether the statement is true or false. Justify your answer. The range of a linear transformation is a vector space.
The statement is true. The range of a linear transformation T, from a vector space V to a vector space W, is a subspace of W.
Determine whether the statement below is true or false. Justify the answer. If B is the standard basis for ℝn, then the B-coordinate vector of an x in ℝn is x itself.
The statement is true. The standard basis consists of the columns of the n×n identity matrix. So [x]B = x1e1 + ••• + xnen = x.
Determine if the given set is a subspace of ℙ6. Justify your answer. All polynomials of degree at most 6, with rational numbers as coefficients.
The zero vector of ℙ6 is in the set because zero is a rational number. The set is closed under vector addition because the sum of two rational numbers is a rational number. The set is not closed under multiplication by scalars because the product of a scalar and a rational number is not necessarily a rational number. The set is not a subspace of ℙ6
Determine if the given set is a subspace of ℙ8. Justify your answer: All polynomials of degree at most 8, with positive real numbers as coefficients.
The zero vector of ℙ8 is not in the set because zero is not a positive real number. The set is closed under vector addition because the sum of two positive real numbers is a positive real number. The set is not closed under multiplication by scalars because the product of a scalar and a positive real number is not necessarily a positive real number. The set is not a subspace of ℙ8
Let V be a nonzero finite-dimensional vector space, and the vectors listed belong to V. Determine whether the statement below is true or false. Justify the answer. If dim V=p, then there exists a spanning set of p+1 vectors in V.
This statement is true. The spanning set of p+1 can be found by taking any spanning set, which contains p vectors, for V, and adjoining the zero vector to it.
Suppose an 8×10 matrix A has eight pivot columns. Is Col A=ℝ8?
Yes. Since A has eight pivot columns, dim Col A=8. Thus, Col A is an eight-dimensional subspace of ℝ8, so Col A is equal to ℝ8.
Let B be the standard basis of the space ℙ2 of polynomials. Use coordinate vectors to test whether the following set of polynomials span ℙ2. Justify your conclusion. −2t+t2, 1+8t−2t2, −5−4t−2t2, −1−13t
Yes; since the matrix whose columns are the B-coordinate vectors of each polynomial has a pivot position in eachrow, the set of coordinate vectors spans ℝ3. By isomorphism between ℝ3 and ℙ2, the set of polynomials spans ℙ2.
If the nullity of a 4×8 matrix A is 4, what are the dimensions of the column and row spaces of A?
dim Col A = 4 dim Row A = 4
