Example: Let V be a vector space over a field F. Let U, W be subspaces of V, neither of which contain the other. Then the union of U and U' is not a subspace.

Proof: Suppose, by way of contradiction, that the union of U and U' is a subspace.
By assumption, there exists a vector u in U which is not in U', and a vector u' in U' which is not in U. Let v = u + u'. Then v does not belong to U, because, if it did, u' = v-u would belong to U, contradicting our assumption. Symmetric reasoning shows that v does not belong to U'.
We have exhibited two elements (u and u') of the union of U and U', whose sum (namely, v) does not belong to the union. Therefore, it is not a subspace.

• Axler Chapter 1, problems 6, 7, 13, 15 (We will cover dimensions on Monday).
• Axler, Chapter 2, problem 1, 6.
  
• Let V be a vector space over the field R of real numbers. Prove that V is not equal to the union of a finite number of proper subspaces. (A subspace is called proper if it is not equal to V.)
• Let A be a list of vectors in a vector space V. Show that span(A) is the intersection of all subspaces containing A.
• (Extra credit) Can a vector space be the union of three proper subspaces?

• Axler Chapter 2, problems 8, 11,13, 14, 15.
• Axler Chapter 3, problems 1, 3.
• (Katznelson 1.2.7). Suppose that U, W are subspaces of a vector space V so that U intersects W only in the trivial vector. Suppose that u1, u2, ... un is a linearly independent list in U, and w1, ..., wm is a linearly independent list in W. Show that u1, u2, ..., un, w1, ..., wm is linearly independent in V.
• (Katznelson 1.2.9; previously listed incorrectly as Katznelson1.2.8) Suppose that V is a finite dimensional vector space. Show that every subspace W of V satisfies dim(W) <= dim(V), and that equality dim(W) = dim(V) holds only when W=V.
• Let V be a finite-dimensional vector space over the complex numbers. Let V' be V BUT considered as a vector space over the real numbers. Show that the dimension of V' is twice the dimension of V.
• (Extra credit). Suppose that we are given a system of linear equations all of whose coefficients are integers (example: 2x+3y+4z=2, x-z=8) and you know that it has a solution in real numbers. Prove that it has a solution in rational numbers.

• Axler Chapter 3, problem 4,6, 12, 14, 15, 16, 23, 25.
• We denote by R the real numbers and Q the rational numbers. Let x in R. Prove that the following two statements are equivalent:
  • x is algebraic, i.e. , satisfies a polynomial equation with rational coefficients.
  • Consider R, the real numbers, as a Q-vector space. The subspace spanned by the infinite list 1, x, x^2, x^3, ... is finite-dimensional.
• Let V be an n-dimensional vector space over a field, and let T: V --> V be a linear map. Let T^k denote the kth power of T (i.e., T^2 =T.T = T composed with T, T^3 = T. T^2 and so on). Suppose T^k = 0 for some positive integer k. Show that T^n = 0. Give an example to show that T^(n-1) need not be zero.

1. Describe (with proof) how the adjoint of a composition of two linear maps relates to the adjoints of each linear map separately.
2. Suppose X, Y are two vector spaces. Let Z be the direct sum of X and Y. Prove that Z* is isomorphic to the direct sum of X* and Y*. (Note: do not assume the vector spaces are finite dimensional.)
3. In the following questions, V, W are finite dimensional vector spaces over the field F, T: V-->W is a linear map and T^: W*-->V* be the adjoint. Prove the following.
   • There exists a basis e_1, ..., e_n for V and a basis f_1, ... f_m for W so that T(e_i) is either equal to f_i, or equal to zero. Describe the adjoint T^ with respect to the dual bases e_i*, f_j*.
   • If T is surjective, then T^ is injective.
   • If T is injective, then T^ is surjective.
   • The dimension of the image of T and the dimension of the image of T^ are the same.
4. Suppose that V is a finite-dimensional vector space with basis e1, ... en. Let e1*, ..., en* be the dual basis for V*.

   Suppose that T: V-->V* has a matrix a_{ij} (with respect to these bases) that is symmetric, i.e., a_{ij} = a_{ji}. Prove that this is "independent of basis", i.e., given any other basis f1, ..., fn, with dual basis f1*, ..., fn*, the matrix of T with respect to (f1, ..., fn) and (f1*, ..., fn*) is also symmetric.

   (Hint: the elegant way to solve ths problem is to find a way to phrase "symmetric" without reference to a basis.)

1. Suppose that M is a square upper triangular matrix. (This means that all entries of M strictly below the diagonal are zero.) Prove, from the definition of determinants as presented in class, that the determinant of M is the product of the diagonal entries.
2. Let D be a nonzero alternating 3-form on R^3. Describe in geometric terms when, for v1, v2, v3 in R^3, the sign of D(v1, v2, v3) is positive.
3. Suppose that D is a nonzero alternating n-form on an n-dimensional vector space V. Suppose that e1, ..., en is a basis for V.
   • Prove that, if we replace e1 by e1 + alpha e_j, for any alpha in F and any j > 1, the value of D(e1, ..., en) remains unchanged.
   • Prove that, if M is an nxn square matrix, then adding any multiple of a row of M to some other row leaves det(M) unchanged. (This gives an efficient way of computing the determinant; if you have not seen determinants before you should compute some determinants of 3x3 matrices using this property and Problem 1.)
4. (This is Katznelson Corollary 4.4.4; try to prove it yourself before looking it up.) Suppose that V is the interior direct sum of subspaces W1, W2. Suppose T: V --> V "preserves" W1 and W2. This means that whenever w in W1, then also T(w) in W1, and similarly for W2. Let T1 = T restricted to W1, and let T2 = T restricted to W2. Prove that det(T) = det(T1) det(T2).
5. Do either Katznelson IV. 5.5. (Hint: the prior problem may be helpful) or Katznelson IV. 5. 12. (The problem about Vandermonde matrices.)
6. Suppose that M is a 2x2 matrix with entries a,b,c,d. Suppose that, for every scalar x in F, the determinant of xM + Id (where Id is the 2x2 identity matrix) equals 1. Prove that M^2 = 0.
7. Extra credit: generalize the previous question to nxn matrices (partial credit for 3x3).
8. Extra credit: Let A: V --> V be a linear transformation on a finite dimensional space V. Let A^: V* ---> V* be the adjoint. Prove det(A) = det(A^). Solutions.

• Axler Chapter 5, problems 4, 5, 6, 7, 11, 12, 21, 23. (For problem 5, you can look up the definition of "invariant" in the first section of Axler Chapter 5).
• Suppose that the characteristic polynomial of a 3x3 matrix M is det(x. Id - M) = (x-1)(x-2)(x-3). What is the determinant of M? What is the characteristic polynomial of M^100?
  Solutions