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Linear Algebra Problems
Math 504 – 505 Jerry L. Kazdan
Topics
1 Basics 2 Linear Equations 3 Linear Maps 4 Rank One Matrices 5 Algebra of Matrices 6 Eigenvalues and Eigenvectors 7 Inner Products and Quadratic Forms 8 Norms and Metrics 9 Projections and Reflections 10 Similar Matrices 11 Symmetric and Self-adjoint Maps 12 Orthogonal and Unitary Maps 13 Normal Matrices
14 Symplectic Maps 15 Differential Equations 16 Least Squares 17 Markov Chains 18 The Exponential Map 19 Jordan Form 20 Derivatives of Matrices 21 Tridiagonal Matrices 22 Block Matrices 23 Interpolation 24 Dependence on Parameters 25 Miscellaneous Problems
The level of difficulty of these problems varies wildly. Some are entirely appropriate for a high school course. Others definitely inappropriate.
Although problems are categorized by topics, this should not be taken very seriously. Many problems fit equally well in several different topics.
Note: To make this collection more stable no new problems will be added in the future. Of course corrections and clarifications will be inserted. Corrections and comments are welcome. Email: [email protected]
I have never formally written solutions to these problems. However, I have frequently used some in Homework and Exams in my own linear algebra courses – in which I often have written solutions. See my web page: https://www.math.upenn.edu/~kazdan/
Notation: We occasionally write M (n, F) for the ring of all n × n matrices over the field F, where F is either R or C. For a real matrix A we sometimes use that the adjoint A∗^ is the transpose and write AT^.
1 Basics
- At noon the minute and hour hands of a clock coincide. a) What in the first time, T 1 , when they are perpendicular? b) What is the next time, T 2 , when they again coincide?
- Which of the following sets are linear spaces?
a) {X = (x 1 , x 2 , x 3 ) in R^3 with the property x 1 − 2 x 3 = 0} b) The set of solutions ~x of A~x = 0, where A is an m × n matrix. c) The set of 2 × 2 matrices A with det(A) = 0. d) The set of polynomials p(x) with
− 1 p(x)^ dx^ = 0. e) The set of solutions y = y(t) of y′′^ + 4y′^ + y = 0. f) The set of solutions y = y(t) of y′′^ + 4y′^ + y = 7e^2 t^. g) Let Sf be the set of solutions u(t) of the differential equation u′′^ − xu = f (x). For which continuous functions f is Sf a linear space? Why? [Note: You are not being asked to actually solve this differential equation.]
- Which of the following sets of vectors are bases for R^2?
a). {(0, 1), (1, 1)} b). {(1, 0), (0, 1), (1, 1)} c). {(1, 0), (− 1 , 0 }
d). {(1, 1), (1, −1)} e). {((1, 1), (2, 2)} f). {(1, 2)}
- For which real numbers x do the vectors: (x, 1 , 1 , 1), (1, x, 1 , 1), (1, 1 , x, 1), (1, 1 , 1 , x) not form a basis of R^4? For each of the values of x that you find, what is the dimension of the subspace of R^4 that they span?
- Let C(R) be the linear space of all continuous functions from R to R.
a) Let Sc be the set of differentiable functions u(x) that satisfy the differential equa- tion u′^ = 2xu + c for all real x. For which value(s) of the real constant c is this set a linear subspace of C(R)? b) Let C^2 (R) be the linear space of all functions from R to R that have two continuous derivatives and let Sf be the set of solutions u(x) ∈ C^2 (R) of the differential equation u′′^ + u = f (x) for all real x. For which polynomials f (x) is the set Sf a linear subspace of C(R)? c) Let A and B be linear spaces and L : A → B be a linear map. For which vectors y ∈ B is the set Sy := {x ∈ A | Lx = y} a linear space?
a) The columns of A are linearly independent. b) The columns of A span Rn^. c) The rows of A are linearly independent. d) The kernel of A is 0. e) The only solution of the homogeneous equations Ax = 0 is x = 0. f) The linear transformation TA : Rn^ → Rn^ defined by A is 1-1. g) The linear transformation TA : Rn^ → Rn^ defined by A is onto. h) The rank of A is n. i) The adjoint, A∗^ , is invertible. j) det A 6 = 0.
- Call a subset S of a vector space V a spanning set if Span(S) = V. Suppose that T : V → W is a linear map of vector spaces. a) Prove that a linear map T is 1-1 if and only if T sends linearly independent sets to linearly independent sets. b) Prove that T is onto if and only if T sends spanning sets to spanning sets.
2 Linear Equations
- Solve the given system – or show that no solution exists:
x + 2y = 1 3 x + 2y + 4 z = 7 − 2 x + y − 2 z = − 1
- Say you have k linear algebraic equations in n variables; in matrix form we write AX = Y. Give a proof or counterexample for each of the following. a) If n = k there is always at most one solution. b) If n > k you can always solve AX = Y. c) If n > k the nullspace of A has dimension greater than zero. d) If n < k then for some Y there is no solution of AX = Y. e) If n < k the only solution of AX = 0 is X = 0.
- Let A : Rn^ → Rk^ be a linear map. Show that the following are equivalent.
a) For every y ∈ Rk^ the equation Ax = y has at most one solution. b) A is injective (hence n ≤ k ). [injective means one-to-one] c) dim ker(A) = 0. d) A∗^ is surjective (onto). e) The columns of A are linearly independent.
- Let A : Rn^ → Rk^ be a linear map. Show that the following are equivalent.
a) For every y ∈ Rk^ the equation Ax = y has at least one solution. b) A is surjective (hence n ≥ k ). [surjective means onto] c) dim im(A) = k. d) A∗^ is injective (one-to-one). e) The columns of A span Rk^.
- Let A be a 4 × 4 matrix with determinant 7. Give a proof or counterexample for each of the following. a) For some vector b the equation Ax = b has exactly one solution. b) For some vector b the equation Ax = b has infinitely many solutions. c) For some vector b the equation Ax = b has no solution. d) For all vectors b the equation Ax = b has at least one solution.
- Let A : Rn^ → Rk^ be a real matrix, not necessarily square.
a) If two rows of A are the same, show that A is not onto by finding a vector y = (y 1 ,... , yk) that is not in the image of A. [Hint: This is a mental computation if you write out the equations Ax = y explicitly.] b) What if A : Cn^ → Ck^ is a complex matrix? c) More generally, if the rows of A are linearly dependent, show that it is not onto.
- Let A : Rn^ → Rk^ be a real matrix, not necessarily square.
a) If two columns of A are the same, show that A is not one-to-one by finding a vector x = (x 1 ,... , xn) that is in the nullspace of A. b) More generally, if the columns of A are linearly dependent, show that A is not one-to-one.
- Let A and B be n × n matrices with AB = 0. Give a proof or counterexample for each of the following.
a) Find the general solution Z of the homogeneous equation AZ = 0.
b) Find some solution of AX =
c) Find the general solution of the equation in part b).
d) Find some solution of AX =
and of AX =
e) Find some solution of AX =
f) Find some solution of AX =
. [Note: ( 72 ) = ( 12 ) + 2 ( 30 )].
[Remark: After you have done parts a), b) and e), it is possible immediately to write the solutions to the remaining parts.]
- Consider the system of equations
x + y − z = a x − y + 2z = b 3 x + y = c
a) Find the general solution of the homogeneous equation. b) If a = 1, b = 2, and c = 4, then a particular solution of the inhomogeneous equa- tions is x = 1, y = 1, z = 1. Find the most general solution of these inhomogeneous equations. c) If a = 1, b = 2, and c = 3, show these equations have no solution. d) If a = 0, b = 0, c = 1, show the equations have no solution. [Note:
0 1
2 4
2 3
].
e) Let A =
. Find a basis for ker(A) and image (A).
- Let A be a square matrix with integer elements. For each of the following give a proof or counterexample. a) If det(A) = ±1, then for any vector y with integer elements there is a vector x with integer elements that solves Ax = y. b) If det(A) = 2, then for any vector y with even integer elements there is a vector x with integer elements that solves Ax = y.
c) If all of the elements of A are positive integers and det(A) = +1, then given any vector y with non-negative integer elements there is a vector x with non-negative integer elements that solves Ax = y. d) If the elements of A are rational numbers and det(A) 6 = 0, then for any vector y with rational elements there is a vector x with rational elements that solves Ax = y.
3 Linear Maps
- a) Find a 2 × 2 matrix that rotates the plane by +45 degrees (+45 degrees means 45 degrees counterclockwise). b) Find a 2 × 2 matrix that rotates the plane by +45 degrees followed by a reflection across the horizontal axis. c) Find a 2 × 2 matrix that reflects across the horizontal axis followed by a rotation the plane by +45 degrees. d) Find a matrix that rotates the plane through +60 degrees, keeping the origin fixed. e) Find the inverse of each of these maps.
- a) Find a 3 × 3 matrix that acts on R^3 as follows: it keeps the x 1 axis fixed but rotates the x 2 x 3 plane by 60 degrees. b) Find a 3 × 3 matrix A mapping R^3 → R^3 that rotates the x 1 x 3 plane by 60 degrees and leaves the x 2 axis fixed.
- Consider the homogeneous linear system Ax = 0 where
A =
Identify which of the following statements are correct? a) Ax = 0 has no solution. b) dim ker A = 2 c) Ax = 0 has a unique solution. d) For any vector b ∈ R^3 the equation Ax = b has at least one solution.
- Find a real 2 × 2 matrix A (other than A = I ) such that A^5 = I.
Show that the square has no other symmetries. Also, show that SR = G, SR^2 = T , and SR^3 = M. f) Investigate the symmetries of an equilateral triangle in the plane. [See https://en.wikipedia.org/wiki/Dihedral_group for more on the symme- tries of regular polygons by the valuable device of representing the symmetries as matrices. See also: https://chem.libretexts.org/Textbook_Maps/Physical_and_Theoretical_Chemistry_Textbook_Maps/Map%3A_Symmetry_ (Vallance)
- Give a proof or counterexample the following. In each case your answers should be brief. a) Suppose that u, v and w are vectors in a vector space V and T : V → W is a linear map. If u, v and w are linearly dependent, is it true that T (u), T (v) and T (w) are linearly dependent? Why? b) If T : R^6 → R^4 is a linear map, is it possible that the nullspace of T is one dimensional?
- Identify which of the following collections of matrices form a linear subspace in the linear space Mat (^2) × 2 (R) of all 2 × 2 real matrices? a) All invertible matrices. b) All matrices that satisfy A^2 = 0. c) All anti-symmetric matrices, that is, AT^ = −A. d) Let B be a fixed matrix and B the set of matrices with the property that AT^ B = BAT^.
- Identify which of the following collections of matrices form a linear subspace in the linear space Mat (^3) × 3 (R) of all 3 × 3 real matrices? a) All matrices of rank 1. b) All matrices satisfying 2A − AT^ = 0.
c) All matrices that satisfy A
- Let V be a vector space and
: V → R be a linear map. If z ∈ V is not in the nullspace of, show that every x ∈ V can be decomposed uniquely as x = v + cz , where v is in the nullspace of ` and c is a scalar. [Moral: The nullspace of a linear functional has codimension one.]
- For each of the following, answer TRUE or FALSE. If the statement is false in even a single instance, then the answer is FALSE. There is no need to justify your answers to this problem – but you should know either a proof or a counterexample. a) If A is an invertible 4 × 4 matrix, then (AT^ )−^1 = (A−^1 )T^ , where AT^ denotes the transpose of A. b) If A and B are 3 × 3 matrices, with rank(A) = rank(B) = 2, then rank(AB) = 2. c) If A and B are invertible 3 × 3 matrices, then A + B is invertible. d) If A is an n × n matrix with rank less than n, then for any vector b the equation Ax = b has an infinite number of solutions. e) ) If A is an invertible 3 × 3 matrix and λ is an eigenvalue of A, then 1/λ is an eigenvalue of A−^1 ,
- For each of the following, answer TRUE or FALSE. If the statement is false in even a single instance, then the answer is FALSE. There is no need to justify your answers to this problem – but you should know either a proof or a counterexample. a) If A and B are 4 × 4 matrices such that rank (AB) = 3, then rank (BA) < 4. b) If A is a 5 × 3 matrix with rank (A) = 2, then for every vector b ∈ R^5 the equation Ax = b will have at least one solution. c) If A is a 4 × 7 matrix, then A and AT^ have the same rank. d) Let A and B 6 = 0 be 2 × 2 matrices. If AB = 0, then A must be the zero matrix.
- Let A : R^3 → R^2 and B : R^2 → R^3 , so BA : R^3 → R^3 and AB : R^2 → R^2.
a) Show that BA can not be invertible. b) Give an example showing that AB might be invertible (in this case it usually is).
- Let A, B , and C be n × n matrices.
a) If A^2 is invertible, show that A is invertible. [Note: You cannot naively use the formula (AB)−^1 = B−^1 A−^1 because it pre- sumes you already know that both A and B are invertible. For non-square matrices, it is possible for AB to be invertible while neither A nor B are (see the last part of the previous Problem 41).] b) Generalization. If AB is invertible, show that both A and B are invertible. If ABC is invertible, show that A, B , and C are also invertible.
- Let A be a real square matrix satisfying A^17 = 0.
a) Show that the matrix I − A is invertible. b) If B is an invertible matrix, is B − A also invertible? Proof or counterexample.
- Linear maps F (X) = AX , where A is a matrix, have the property that F (0) = A0 = 0, so they necessarily leave the origin fixed. It is simple to extend this to include a translation, F (X) = V + AX, where V is a vector. Note that F (0) = V. Find the vector V and the matrix A that describe each of the following mappings [here the light blue F is mapped to the dark red F ].
a). b).
c). d).
- Find all linear maps L : R^3 → R^3 whose kernel is exactly the plane { (x 1 , x 2 , x 3 ) ∈ R^3 | x 1 + 2x 2 − x 3 = 0 }.
- Let A be a matrix, not necessarily square. Say V and W are particular solutions of the equations AV = Y 1 and AW = Y 2 , respectively, while Z 6 = 0 is a solution of the homogeneous equation AZ = 0. Answer the following in terms of V , W , and Z. a) Find some solution of AX = 3Y 1. b) Find some solution of AX = − 5 Y 2. c) Find some solution of AX = 3Y 1 − 5 Y 2. d) Find another solution (other than Z and 0) of the homogeneous equation AX = 0. e) Find two solutions of AX = Y 1. f) Find another solution of AX = 3Y 1 − 5 Y 2. g) If A is a square matrix, then det A =? h) If A is a square matrix, for any given vector W can one always find at least one solution of AX = W? Why?
- Let V be an n-dimensional vector space and T : V → V a linear transformation such that the image and kernel of T are identical. a) Prove that n is even. b) Give an example of such a linear transformation T.
- Let V, W be two-dimensional real vector spaces, and let f 1 ,... , f 5 be linear transfor- mations from V to W. Show that there exist real numbers a 1 ,... , a 5 , not all zero, such that a 1 f 1 + · · · + a 5 f 5 is the zero transformation.
- Let V ⊂ R^11 be a linear subspace of dimension 4 and consider the family A of all linear maps L : R^11 − > R^9 each of whose nullspace contain V. Show that A is a linear space and compute its dimension.
- Let L be a 2 × 2 matrix. For each of the following give a proof or counterexample.
a) If L^2 = 0 then L = 0. b) If L^2 = L then either L = 0 or L = I. c) If L^2 = I then either L = I or L = −I.
- Find all four 2 × 2 diagonal matrices A that have the property A^2 = I.
Geometrically interpret each of these examples as linear maps.
a) Find a basis for this space. b) Let D : P 2 → P 2 be the derivative operator D = d/dx. Using the basis you picked in the previous part, write D as a matrix. Compute D^3 in this situation. Why should you have predicted this without computation?
- Let P 3 be the space of polynomials of degree at most 3 anD let D : P 3 → P 3 be the derivative operator. a) Using the basis e 1 = 1, e 2 = x, e 3 = x^2 , 4 = x^3 find the matrix De representing D. b) Using the basis 1 = x^3 , 2 = x^2 , 3 = x, 4 = 1 find the matrix D representing D. c) Show that the matrices De and D are similar by finding an invertible map S : P 3 → P 3 with the property that D = SDeS−^1.
- a) Let {e 1 , e 2 ,... , en} be the standard basis in Rn^ and let {v 1 , v 2 ,... , vn} be another basis in Rn^. Find a matrix A that maps the standard basis to this other basis. b) Let {w 1 , w 2 ,... , wn} be yet another basis for Rn^. Find a matrix that maps the {vj } basis to the {wj } basis. Write this matrix explicitly if both bases are orthonormal.
- Consider the two linear transformations on the vector space V = Rn^ :
R = right shift: (x 1 ,... , xn) → (0, x 1 ,... , xn− 1 ) L = left shift: (x 1 ,... , xn) → (x 2 ,... , xn, 0). Let A ⊂ End (V ) be the real algebra generated by R and L. Find the dimension of A considered as a real vector space.
- Let S ⊂ R^3 be the subspace spanned by the two vectors v 1 = (1, − 1 , 0) and v 2 = (1, − 1 , 1) and let T be the orthogonal complement of S (so T consists of all the vectors orthogonal to S ). a) Find an orthogonal basis for S and use it to find the 3 × 3 matrix P that projects vectors orthogonally into S. b) Find an orthogonal basis for T and use it to find the 3 × 3 matrix Q that projects vectors orthogonally into T. c) Verify that P = I − Q. How could you have seen this in advance?
- Given a unit vector w ∈ Rn^ , let W = span {w} and consider the linear map T : Rn^ → Rn^ defined by T (x) = 2 ProjW (x) − x, where ProjW (x) is the orthogonal projection onto W. Show that T is one-to-one.
- [The Cross Product as a Matrix]
a) Let v := (a, b, c) and x := (x, y, z) be any vectors in R^3. Viewed as column vectors, find a 3 × 3 matrix Av so that the cross product v × x = Avx. Answer:
v × x = Avx =
0 −c b c 0 −a −b a 0
x y z
where the anti-symmetric matrix Av is defined by the above formula. b) From this, one has v × (v × x) = Av(v × x) = A^2 vx (why?). Combined with the cross product identity u × (v × w) = 〈u, w〉v − 〈u, v〉w , show that
A^2 vx = 〈v, x〉v − ‖v‖^2 x.
c) If n = (a, b, c) is a unit vector, use this formula to show that (perhaps surprisingly) the orthogonal projection of x into the plane perpendicular to n is given by
x − (x · n)n = −A^2 nx = −
−b^2 − c^2 ab ac ab −a^2 − c^2 bc ac bc −a^2 − b^2
(^) x
(See also Problems 193, 233, 234, 235, 273).
- Let V be a vector space with dim V = 10 and let L : V → V be a linear transformation. Consider Lk^ : V → V , k = 1, 2 , 3 ,.. .. Let rk = dim(Im Lk), that is, rk is the dimension of the image of Lk^ , k = 1, 2 ,.. .. Give an example of a linear transformation L : V → V (or show that there is no such transformation) for which: a) (r 1 , r 2 ,.. .) = (10, 9 ,.. .); b) (r 1 , r 2 ,.. .) = (8, 5 ,.. .); c) (r 1 , r 2 ,.. .) = (8, 6 , 4 , 4 ,.. .).
- Let S be the linear space of infinite sequences of real numbers x := (x 1 , x 2 ,.. .). Define the linear map L : S → S by
Lx := (x 1 + x 2 , x 2 + x 3 , x 3 + x 4 ,.. .).
a) Find a basis for the nullspace of L. What is its dimension? b) What is the image of L? Justify your assertion. c) Compute the eigenvalues of L and an eigenvector corresponding to each eigenvalue.
- Let A be a real matrix, not necessarily square.
c) If the vector z = (z 1 ,... , zn) satisfies 〈z, w〉 = 0, show that z is an eigenvector with eigenvalue λ = 0. d) If trace (A) 6 = 0, show that λ = trace (A) is an eigenvalue of A. What is the corresponding eigenvector? e) If trace (A) 6 = 0, prove that A is similar to the n × n matrix
c 0... 0 0 0... 0
............ 0 0... 0
where c = trace (A) f) If trace (A) = 1, show that A is a projection, that is, A^2 = A. g) What can you say if trace (A) = 0? h) Show that det(A + I) = 1 + det A.
- Let A be the rank one n × n matrix A = (vivj ), where ~v := (v 1 ,... , vn) is a non-zero real vector. a) Find its eigenvalues and eigenvectors. b) Find the eigenvalues and eigenvectors for A + cI , where c ∈ R. c) Find a formula for (I + A)−^1. [Answer: (I + A)−^1 = I − (^) 1+‖^1 ~v‖ 2 A.]
- [Generalization of Problem 79(b)] Let W be a linear space with an inner product and A : W → W be a linear map whose image is one dimensional (so in the case of matrices, it has rank one). Let ~v 6 = 0 be in the image of A, so it is a basis for the image. If 〈~v, (I + A)~v〉 6 = 0, show that I + A is invertible by finding a formula for the inverse.
Answer: The solution of (I + A)~x = ~y is ~x = ~y −
‖~v‖^2 ‖~v‖^2 + 〈~v, A~v〉 A~y so
(I + A)−^1 = I −
‖~v‖^2 ‖~v‖^2 + 〈~v, A~v〉
A.
5 Algebra of Matrices
- Which of the following are not a basis for the vector space of all symmetric 2 × 2 matrices? Why?
a)
b)
c)
d)
e)
f)
- For each of the sets S below, determine if it is a linear subspace of the given real vector space V. If it is a subspace, write down a basis for it. a) V = Mat 3 × 3 (R), S = {A ∈ V | rank(A) = 3}. b) V = Mat 2 × 2 (R), S = {
( (^) a b c d
∈ V | a + d = 0}.
- Every real upper triangular n×n matrix (aij ) with aii = 1, i = 1, 2 ,... , n is invertible. Proof or counterexample.
- Let L : V → V be a linear map on a vector space V.
a) Show that ker L ⊂ ker L^2 and, more generally, ker Lk^ ⊂ ker Lk+1^ for all k ≥ 1. b) If ker Lj^ = ker Lj+1^ for some integer j , show that ker Lk^ = ker Lk+1^ for all k ≥ j. Does your proof require that V is finite dimensional? c) Let A be an n × n matrix. If Aj^ = 0 for some integer j (perhaps j > n), show that An^ = 0.
- Let L : V → V be a linear map on a vector space V and z ∈ V a vector with the property that Lk−^1 z 6 = 0 but Lkz = 0. Show that z , Lz ,... Lk−^1 z are linearly independent.
- Let A, B , and C be any n × n matrices.
a) Show that trace(AB) = trace(BA). b) Show that trace(ABC) = trace(CAB) = trace(BCA). c) trace(ABC) = trace(? BAC). Proof or counterexample.
