Math 104X.01 Homework Assignments

Day 01: 11 January

1. Reading: Sections 1.1-1.3
2. Topics: Linear equations and Gaussian elimination
3. Homework Exercises:
   • Section 1.2: 2, 3*, 5, 8, 10*
   • Section 1.3: 1, 2, 6*, 10, 12*, 18*

Day 02: 16 January

1. Reading: Section 1.4
2. Topics: Matrices and matrix multiplication
3. Homework Exercises:
   • Section 1.4: 1, 6, 7, 12, 13*, 17*, 20, 24, 29*, 31, 34*

Day 03: 18 January

1. Reading: Sections 1.5-1.6
2. Topics: A = LU factorization, inverses and transposes
3. Homework Exercises:
   • Section 1.5: 4, 5, 6*, 9, 11, 15*, 19, 27, 28*
   • Section 1.6: 1, 2*, 6*, 10

Day 04: 23 January

1. Reading: Sections 2.1-2.3
2. Topics: Vector spaces and subspaces
3. Homework Exercises:
   • Section 2.1: 1, 2*, 4, 5*, 8, 10, 11, 12, 18*, 20*

Day 05: 25 January

1. Reading: Section 2.4
2. Topics: Row Echelon form, linear (in)dependence, basis, dimension
3. Homework Exercises:
   • Section 2.1: 21, 22*, 25
   • Section 2.2: 2, 3, 6*, 7, 12*, 13, 18*, 25*
   • Section 2.3: 1, 5, 7, 9*, 22*, 29, 37*

Day 06: 30 January

1. Reading: Sections 2.5-2.6
2. Topics: The Fundamental Subspaces, Networks, Kirchoff's Law, Linear Transformations
3. Homework Exercises:
   • Section 2.4: 2, 3*, 5, 7, 9, 13, 14*, 21, 27*, 36*, 37*

Day 07: 01 February

1. Reading: Section 2.6 (again! 2.6 is VERY important)
2. Topics: Linear Transformations
3. Homework Exercises:
   • Section 2.6: 1, 4*, 6, 7*, 14, 18*, 25, 27, 28, 30*, 32, 34*, 36

Day 08: 06 February

1. Reading: Section 2.6 (again! 2.6 is VERY important)
2. Topics: Matrices associated to linear transformations

Day 09: 08 February

1. First in-class test. (Covers material from Chapters 1 and 2, but not Section 1.7 or Section 2.5)
2. Here is a .pdf copy of the key to the test.

Day 10: 13 February

1. Reading: Sections 3.1 and 3.2
2. Topics: Orthogonality, Length, Inner products
3. Homework Exercises:
   • Section 3.1: 1, 2, 4*, 8, 12, 19, 21, 25, 26*, 35*, 43*, 45, 46
   • Section 3.2: 3*, 5*, 7, 8*

Day 11: 15 February

1. Reading: Sections 3.3 and 3.4
2. Topics: Projections
3. Homework Exercises:
   • Section 3.2: 9, 10*, 11, 13, 16, 17*
   • Proof*: Let W₁ and W₂ be (finite dimensional) subspaces of a vector space V. Prove that
     dim (W₁ + W₂) = dim(W₁) + dim(W₂) - dim(W₁ ∩ W₂).

Day 12: 20 February

1. Reading: Sections 3.3 and 3.4 (again)
2. Topics: Least Squares, Orthogonal Transformations, Gram-Schmidt
3. Homework Exercises:
   • Section 3.3: 3, 6*, 7, 11*, 12*, 19
   • Section 3.4: 2, 3, 4*, 6*, 11*, 14*, 28

Day 13: 22 February

1. Reading: Section 3.4, esp. pp. 182-185
2. Topics: Inner products, Hilbert Space, Fourier series
3. Homework Exercises: (If your browser doesn't correctly interpret the math html code below, you can download a .pdf file of the homework problems here.
   • Section 3.4: 20*, 21*, 23, 24*
   • Proof*: Let (,) be an inner product on the vector space V. Show that, for any subset S ⊆ V, the set S^⊥, consisting of the vectors x∈V that satisfy (x,y)=0 for all y∈S, is a subspace of V. If S itself is a subspace of V and (,) is non-degenerate, is it always true that (S^⊥)^⊥ = S? (Extra credit: If V is finite dimensional, S is a subspace of V and (,) is a non-degenerate inner product on V, is it always true that (S^⊥)^⊥ = S?)
   • Calculation*: Let V = C([0,2π]) be the continuous functions on the interval [0,2π] and let the inner product on V be the one we defined in class: (f,g) = ∫₀^2π f(x)g(x) dx. Verify that the functions
     v₀ = 1, v₁ = sin x, v₂ = cos x, … v_2k-1 = sin kx, v_2k = cos kx, … are orthogonal with respect to this inner product. Let W_k ⊂ V be the subspace (of dimension 2k+1) spanned by v₀,…v_2k and let P_k: V → W_k be the orthogonal projection onto W_k. Apply Plancherel's Theorem
     ||f||² = lim_k→∞ ||P_k(f)||²
   to the function f(x) = x - π and derive Euler's famous formula for the sum of the reciprocals of the squares.

Day 14: 27 February

1. Reading: Sections 4.1 and 4.2
2. Topics: Determinants!
3. Homework Exercises:
   • Section 4.2: 1, 2*, 7, 10*, 12*, 13, 27, 34*, 35

Day 15: 01 March

1. Reading: Sections 4.3 and 4.4
2. Topics: More Determinants! (and examples)
3. Homework Exercises:
   • Section 4.3: 1, 5*, 6, 8*, 11, 22*, 23, 28*, 31, 34*
   • Section 4.4: 1, 2*, 3, 13*, 21, 22*
   • Proof*: Let u and v be any two vectors in Rⁿ. Prove that det( I_n + u v^T ) = 1 + v^T u.
   (Hint: There are at least two ways to do this: Brute force is a little messy but can be made to work, either by using row or column expansion or reduction, or the Big Formula. Another way is to observe that it is true if u = 0 (obvious) or u = e₁ (why?) and then find a way to reduce the general case when u is not zero to the case u = e₁.)

Day 16: 06 March

Day 17: 08 March

1. Second in-class test. (Covers material from Chapters 3 and 4)
2. Here is a .pdf copy of the key to the test.

Day 18: 20 March

Day 19: 22 March

Day 20: 27 March

Day 21: 29 March

Day 22: 03 April

Day 23: 05 April

Day 24: 10 April

Day 25: 12 April

1. Third in-class test?

Day 26: 17 April

Day 27: 19 April

Day 28: 24 April