A brief survey of linear algebra

{\LARGE\bf A brief survey of linear algebra}\newline \newline \newline

A brief survey of linear algebra


Tom Carter



http://cogs.csustan.edu/~tom/linear-algebra

Santa Fe Institute
Complex Systems Summer School

June, 2001

Our general topics: Top

(ex): exercises.

Why linear algebra Top

Vector spaces Top

Exercises: Vector spaces Top

  1. Using the basic properties listed above, prove the additional properties:

    1. 0 * v = 0
    2. a * 0 = 0
    3. (-1) * v = -v
    4. -(-v) = v
    5. For a F and v V, av = 0 iff a = 0 or v = 0.

  2. Prove that the additive identity 0 is unique.
  3. Prove that the additive inverse -v is unique.

Examples of vector spaces Top

Exercises: Examples of vector spaces Top

  1. Show that each of the examples listed in this section is a vector space.

  2. Consider the set of points in R2 of the forms (x, 0), x R, and (0, y), y R (i.e., the union of the X-axis and the Y-axis). Show that with the usual vector addition in R2, this is not a vector space over R.

  3. Consider the set R2 with usual vector addition:
    (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2),
    but with ``scalar multiplication'' given by
    a * (x, y) = (ax, y)
    for a R. Show that this is not a vector space over R.

Subspaces Top

Exercises: Subspaces Top

  1. Show that each of the examples identified in this section as a subspace actually is a vector subspace.

  2. Which of the following are vector subspaces of R3?

    1. {(x1, x2, x3) R3 | 3x1 + 2x2 - x3 = 0}
    2. {(x1, x2, x3) R3 | 3x1 + 2x2 - x3 = 4}
    3. {(x1, x2, x3) R3 | x1x2x3 = 0}

  3. Suppose that U is a vector subspace of V, and V is a vector subspace of W. Show that U is a vector subspace of W.

  4. Show that the intersection of any collection of vector subspaces of V is a vector subspace of V.

  5. Let
    l2 = {(ai) R |

    i = 0 
    |ai|2 < }.
    Show that l2 is a vector subspace of R.

  6. Let
    L2 = {f C0(R|


    - 
    |f(x)|2 dx < }.
    Show that L2 is a vector subspace of C0(R).

Linear dependence and independence Top

From here on, we'll assume that U and V are vector spaces over a field F.

Exercises: Linear dependence and independence Top

  1. Verify the statements in each of the six examples in this section.

  2. Suppose that P0, P1, , Pn are polynomials in Fn[x], and Pi(1) = 0 for i = 0, 1, ,n. Show that {P0 , P1, , Pn} is not linearly independent in Fn[x].

  3. In C(R), show that each of these sets of functions is linearly independent:

    1. {xn | n = 0, 1, 2, }.

    2. {sin(nx) | n = 0, 1, 2, }.

    3. {enx | n = 0, 1, 2, }.

Span of a set of vectors Top Exercises: Span of a set of vectors Top

  1. Verify the statements in each of the eight examples in this section.

  2. Show that if span({v0, v1, , vn}) = V, then span({v0 - v1, v1 - v2, ,vn-1 - vn, vn}) = V.

Basis for a vector space Top Exercises: Basis for a vector space Top

  1. Verify the statements in each of the five examples in this section.

  2. Let U be the subspace of R6 given by U = {(x1, x2, x3, x4, x5, x6) R6 |  2x1 + 3x3 = 0 and x2 = 4x5}.

    Find a basis for U.

  3. Show that if U1 and U2 are subspaces of a finite dimensional vector space, then dim(U1 + U2) = dim(U1) + dim(U2) - dim(U1 U2).

Linear transformations Top Exercises: Linear transformations Top

  1. Verify each of the statements marked (ex) in this section.

  2. Verify that each of the 7 examples actually are linear transformations.

  3. Show that the usual calculus derivative [d / dx] : C(R) C(R) given by:
    d f(x)
    dx
    =
    lim
    h 0 
    f(x+h) - f(x)
    h
    is a linear transformation.

  4. Is the usual calculus indefinite integral

    f(x)dx
    a linear transformation? Why or why not? What about the definite integral?

Morphisms - mono, epi, and iso Top

Exercises: Morphisms - mono, epi, and iso Top

  1. Verify each of the statements marked (ex) in this section.

  2. Show that if a function has an inverse, then the inverse is unique.

  3. Consider the function T : Fn[x] Fn+1 given by
    T(a0 + a1x + + anxn) = (a0, , an).
    Show that this function is an isomorphism.

  4. Consider the function T : C matrix(R, 2, 2) given by
    T(a + bi)
    =


    a
    -b
    b
    a


    1. Show that if we consider C as a real vector space, this function is a monomorphism.
    2. Show that this function also respects complex multiplication, that is,
      T((a + bi)(c + di)) = T(a + bi) T(c + di).

  5. More generally, consider the function T : matrix(C, n, n) matrix(R, 2n, 2n) given by
    T







    a11 + b11i
    a1n + b1ni
    :
    :
    an1 + bn1i
    ann + bnni








           =  








    a11
    -b11
    a1n
    -b1n
    b11
    a11
    b1n
    a1n
    :
    :
    :
    :
    an1
    -bn1
    ann
    -bnn
    bn1
    an1
    bnn
    ann









    1. Show that if we consider matrix(C, n, n) as a real vector space, this function is a monomorphism.
    2. Show that this function also respects matrix multiplication, that is,
      T(A B) = T(A) T(B).

  6. What the heck. Let H be the quaternions (described above). Consider the function T : matrix(H, n, n) matrix(C, 2n, 2n) given by
    T







    a11 + b11j
    a1n + b1nj
    :
    :
    an1 + bn1j
    ann + bnnj








           =  












    a11
    -
    b11
     
    a1n
    -
    b1n
     
    b11
    a11
    b1n
    a1n
    :
    :
    :
    :
    an1
    -
    bn1
     
    ann
    -
    bnn
     
    bn1
    an1
    bnn
    ann













    1. Show that if we consider matrix(H, n, n) as a complex vector space, this function is a monomorphism.
    2. Show that this function also respects matrix multiplication, that is,
      T(A B) = T(A) T(B).

    Thus, if we denote by O(n), U(n), and Sp(n) the distance preserving linear operators on Rn, Cn, and Hn respectively (called the orthogonal, unitary, and symplectic groups), then we have the monomorphisms:

    O(n) Sp(n) U(2n) O(4n)
         Sp(4n) U(8n) O(16n)

Linear operators Top Exercises: Linear operators Top

  1. Find a linear operator S such that
    S2
    =


    -1
    0
    0
    -1


  2. If S is the operator
    S
    =









    0
    - p
    4
    p
    4
    0









    ,
    what is exp(S)?

  3. Solve the system of equations:
    3x1
    +
    x2
    -
    2x3
    =
    3
    x1
    -
    2x2
    -
    x3
    =
    1
    2x1
    -
    4x2
    -
    x3
    =
    3
  4. Solve the differential equation:
    d5f
    dx5
    - 7 d4f
    dx4
    + 23 d3f
    dx3
    - 45 d2f
    dx2
    + 48 df
    dx
    -20f
          = 26cos(x) - 18sin(x)
    Hint:
    x5 - 7x4 + 23x3 - 45x2 + 48x - 20
          = (x - 1)(x - 2)2(x2 - 2x + 5)

  5. Find the general solution to the difference equation
    (E2 - E - 1)((an)) = (0)

  6. Find the general solution to the difference equation
    (E4 + 2E2 + 1)((an)) = (0)

  7. Verify that

    1. D, the derivative, is a linear operator.
    2. (D- r)k(xk-1erx) = 0
      (D2 -2sD+ s2 + t2)k(xk-1esxcos(tx))
      =
      0
      (D2 -2sD+ s2 + t2)k(xk-1esxsin(tx))
      =
      0
    3. E, the discrete increment, is a linear operator.
    4. (E - a)k((njan)) = (0) for 0 j < k.

  8. What happens in nonlinear cases? Sometimes they are manageable, sometimes not.

    1. Solve the differential equation
      Df = b * f * (1 - f).

    2. What can be said about the difference equation
      E((an)) = (b * an * (1 - an))?
      Note: this is often called the logistics equation.

Normed linear spaces Top Exercises: Normed linear spaces Top

  1. Verify that each of the examples actually are norms.

  2. In R2, draw the unit circles

    1. circle1(1) = {(x, y) R2 | ||(x, y)||1 = 1}
    2. circle3/2(1) = {(x, y) R2 |                 
      ||(x, y)||3/2 = 1}
    3. circle2(1) = {(x, y) R2 | ||(x, y)||2 = 1}
    4. circle3(1) = {(x, y) R2 | ||(x, y)||3 = 1}
    5. circle(1) = {(x, y) R2 |                 
      ||(x, y)|| = 1}

  3. Suppose that V is a real or complex vector space. An inner product on V is a conjugate-bilinear function on V:
    < ·, · > : V x V F
    where, for all v1, v2, v V, and a F,
    < v , v >
    0
    < v , v >
    =
    0 iff v = 0
    < v1 + v2 , v >
    =
    < v1 , v > + < v2 , v >
    < v1 , v2 >
    =

    < v2 , v1 >
     
    < av1 , v2 >
    =
    a < v1 , v2 > .
    Show that for an inner product,
    < v , v1 + v2 >
    =
    < v , v1 > + < v , v2 >
    < v1 , av2 >
    =

    a
     
    < v1 , v2 > .
    Show that for a finite dimensional real or complex vector space V with basis (v1, v2, , vn), the function f : V x V F given by
    f


    i 
    ai vi,

    i 
    bi vi
    =

    i 
    ai
    bi
     
    is an inner product.
  4. Recall that a linear functional on a vector space V is a linear map f : V F. For a finite dimensional real or complex inner product space V, define the dual space of V to be the space
    V* = { f : V F |  f is a linear functional}
    Show that V* is a vector space over F, and that V and V* are isomorphic to each other.

    Hint: Show that every f V* corresponds with a function of the form

    < vf , · > : V F
    for some vf V.

Eigenvectors and eigenvalues Top Exercises: Eigenvectors and eigenvalues Top

  1. Show that if v is an eigenvector of T corresponding with the eigenvalue l, and a F (a 0), then av is also an eigenvector of T corresponding with l. In particular, eigenvectors are not unique.

  2. Define T L(F2) by T((x, y)) = (y, x). Find all the eigenvalues and eigenvectors of T.

  3. Define T L(F3) by T((x, y, z)) = (-y, 0, 2z). Find all the eigenvalues and eigenvectors of T.

  4. Define T L(F) by T((a1, a2, )) = (a2, a3, ) (i.e., T is the left shift operator). Find all the eigenvalues and eigenvectors of T.

  5. Suppose T L(V) is invertible. Show that l 0 is an eigenvalue of T if and only if l-1 is an eigenvalue of T-1.

  6. Suppose S, T L(V). Show that ST and TS have the same eigenvalues.

  7. Give an example of an operator whose matrix has all zeros on the diagonal with respect to some basis, but which is invertible. Give an example of an operator whose matrix has all diagonal elements non-zero with respect to some basis, but which is singular (i.e., has no inverse).

  8. Show that if S, T L(V), S is invertible, and P(x) F[x], then P(S-1TS) = S-1P(T)S.

  9. Suppose that V is a finite dimensional normed complex vector space, and T is an isometry of V (i.e., ||T(v)|| = ||v|| for all v V). Show that every eigenvalue l of T has |l| = 1. Hint: show that there is a basis for V consisting of eigenvectors ei, with ||ei|| = 1 for 1 i n.

  10. Let V be a complex vector space, and T L(V). Let P(x) C[x]. Show that a C is an eigenvalue of P(T) if and only if a = P(l) for some eigenvalue l of T.

  11. Is the preceding true for real vector spaces? Why not? (Note: this is another example of why C is so much nicer a place to work than R ...)

    Is the preceding true if we replace C[x] with C[x], power series? If so, show it. If not, give a counterexample.

Change of basis Top Exercises: Change of basis Top

  1. Show that if A and B are operators on a finite dimensional vector space with AB = I, the identity operator, then BA = I, and so B = A-1.

  2. Suppose that T is an operator that has the same matrix with respect to every basis. Show that T must be some multiple of the identity operator I.

Trace and determinant Top Exercises: Trace and determinant Top

  1. Given two real or complex n x n matrices A and B, show that tracem(AB) = tracem(BA).

  2. Given an operator T on a real or complex finite dimensional vector space V, show that tracem([T]) and detm([T]) are independent of the basis used for the matrix representation [T]. Hint: use the change of basis formula [T]S1 = [B]-1[T]S2[B], the previous problem, and detm(AB) = detm(BA).

  3. Show that trace is linear, that is, trace(aS + bT) = atrace(S) + btrace(T).

  4. Show that trace(T) = i = 1n[T]ii.

  5. For two operators S and T, show that [S, T] I. Recall that [S,T] = ST - TS.

  6. Show by example that in general, trace(ST) trace(S)trace(T). In particular, find an operator T on a real vector space V with trace(T2) < 0.

  7. Suppose A is an n x n real matrix, S L(Rn) has matrix representation A with respect to some basis, and T L(Cn) has matrix representation A with respect to some basis. Show that trace(S) = trace(T) and det(S) = det(T).

  8. Show that if T is an isometry on a finite dimensional normed vector space, then |det(T)| = 1.

  9. Show that if T is an operator on a complex vector space of dimension n, and a C, then det(aT) = andet(T).

Top

References

[1]
Axler, Sheldon, Linear Algebra Done Right, second edition, Springer-verlag, New York, 1997.

[2]
Finney, Ross L., and Ostberg, Donald R., Elementary Differential Equations with Linear Algebra, Addison-Wesley, 1976.

[3]
Hoffman, Kenneth, and Kunze, Ray, Linear Algebra, second edition, Engineering/Science/Mathematics, 1971.

[4]
Naimark, M. A., Normed Algebras, Wolters-Noordhoff, Groningen, the Netherlands, 1974.

[5]
Noble, Ben, and Daniel, James W., Applied Linear Algebra, third edition, Engineering/Science/Mathematics, 1988.

[6]
Prugovecki, Eduard, Quantum Mechanics in Hilbert Space, second edition, Academic Press, New York, 1981.

[7]
Rudin, Walter, Functional Analysis, McGraw-Hill, New York, 1973.

[8]
Strang, Gilbert, Linear Algebra and its Applications, Academic Press, New York, 1976.

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