Summary of lectures and homework assignments

• 1/28:    Introduction.  We looked at examples of fields and defined a vector space over a field. HW due 2/1
• p. 9:  2,3,5,6,7,8,10,12,14  Some of you do not have books yet, so hand in the following Friday which can be done using class notes. Do the rest later.
• 6. Let  A1 and A2 be vectors in Rⁿ.  Show that the set of all vectors B so that B is perpendicular to both A1 and A2 is a subspace.
• 7. Generalize 6.  Let A₁, A₂, ...,A _r be vectors in Rⁿ.  Let W be the set of vectors B in Rⁿ such that B^.A_i = 0 for every i=1,2,...,r.  Show that W is a subspce of Rⁿ.
• 8. Show that the following sets in R² form subspaces.
• a) The set of (x,y) such that x=y.
• b) The set of (x,y) such that x-y=0.
• c) The set of (x,y) such that x+4y=0.
• 10. Let U and W be subspaces of a vector space V.  Show that the intersection of U and W is a subspace of V.  Show that U+W is a subspace of V.  (Define U+W to be the set of all elements of V of the form u+w where u is in U and w is in W.)
• 12. Let K be the set of all numbers which can be written in the form a + b 2^.5  (that is a + b square root of 2), where a and b are rational numbers.  Show that K is a field.
• 14: Let c be a rational number > 0 and let g be a real number so g² = c.  Show that the set of all numbers which can be written in the form a+bg, for a and b rational, is a field.
• recall that a subset K of the complex numbers is a field if
1.  Whenever u and v are in K then u+v and uv are both in K.
2.  Whenver u is in K then -u is in K, and if u is not 0 then u ^-1 is in K also.
3.  The numbers 0 and 1 are in K.
• recall that a subset S of a vector spce V over K is a subspace if
1.   whenever v and w are in S, then v+w is also in S.
2.   whenever v is in S and c is in K, then cv is in S.
3.    0 is in S.
• 1/30:   We looked at subspaces.  Handout on fields.
• 2/1:   We looked a linear independence and generating sets and bases.
• 2/4:   Bases and dimension.  Homework due 2/8:
• p 14 1af, 2a, 3a, 4, 5a, 7, 8, 10;  p. 22: 1,3 Some of you do not have books yet, so hand in the following Friday which can be done using class notes. Do the rest later.
• 14-1a Show that (1,1,1) and (0,1,-2) are linearly independent (over C or R).
• 14-1f Show that (1,2) and (1,3) are linearly independent (over C or R).
• 14-2a Express X=(1,0) as a linear combination of A=(1,1) and B=(0,1).  Find the coordinates of X with respect to A,B.
• 14-3a: Find the coordinates of (1,0,0) with respect to the basis { (1,1,1), (-1,1,0),   (1,0,-1)}
• 14-4:  Let (a,b) and (c,d) be two vectors in the plane.  If ad-bc=0,  show that they are linearly dependent. If ad-bc is nonzero, show they are linearly independent.
• 14-5a  Consider the vector space of all functions of a variable t. Show that the functions 1 and t are linearly independent.
• 14-7  What are the coordinates of the function f(t) = 3 sin t + 5 cos t with respect to the basis {sin t, cos t}?
• 14-8:  let D be the derivative d/dt.  Let f(t) be as in 7 above. What are the coordinates of the function Df(t) with respect to the basis {sin t, cos t}?
• 14-10 Let v.w be elements of a vector space and assume v is nonzero.  If v,w are linearly dependent, show there is a scalar a such that w=av.
• 22-1 Let V=R² and let W be the subspace generated by (2,1).  let U be the subspace generated by (0,1).  Show that V is the direct sum of W and U. If U' is the subspace generated by (1,1), show that V is also the direct sum of W and U'.
• 22-3:  Let A,B be two vectors in R² and assume neither of them is 0.  If there is no number c so that cA=B, show that A,B form a basis for R² and that R² is the direct sum of the subspaces generated by A and B respectively.
• 2/6:    We looked at direct sums of subspaces and products of vector spaces.
• 2/8:    More direct sums. WS2
• 2/11:  We looked at Matrices. HW due Friday 2/15:
• p. 27 (Matrices)  6, 7, 9, 10
• p. 28 (Dimension) 1, 3, 4, 9
• p. 31 1
• p. 36 2, 4, 7
• 2/13   I forget what we did, probably more matrices.
• 2/15  We looked at functions in general, injectivity, surjectivity.
• 2/18  We looked at linear transformations.WS3. HW due Friday 2/22
• p. 50 5,6,8
• p. 57 1,2,4,7,9,11,14,15,18
• p. 63 5,6
• 2/20 We looked some at linear transformations.  We proved that if S is a linearly independent subset of a vector space V and f:S->W then there is a linear transformation L:V->W so that L(v)=f(v) for every v in S.  As a corollary, there is a linear projection P:V->U to any subspace U of V.  Along the way we showed that any linearly independent set may be enlarged to a basis. Finally we defined a vector space structure on the set L (V,W) of linear transformations of V to W .  We also defined the dual space V* = L(V,K).
• 2/22 We looked at kernel and image of linear transformations.  Showed the dimensions of the kernel and image add up to the dimension of the domain.  WS4.  Responding to the request to get homework assigned earlier, here is the assignment due next Friday 3/1:
• p 65: 16,18
• p 70:1,2,7,10
• p. 87: 1
• p. 93: 1,8
• 2/25  We looked at matrices associated to linear transformations between finite dimensional vector spaces.  In particular, if B and B' are ordered bases of V and W, and L:V->W  is a linear transformation, then we assign an m x n matrix to L.  This is all chapter IV.
• 2/27:  We did some problems.  We looked at change of basis matrix.
• 3/1: More chapter 4.
• 3/4: We showed among other things that any linear map  T:V->W between finite dimensional vector spaces has the form [I 0; 0 0] with respect to some bases of V and W.  That is there is an identity matrix in the upper left corner and the rest zeros.  The size of the identity matrix is the dimension of Im(T).
• 3/6: Exam #1
• 3/8:  We looked at scalar products- showed that any scalar product on a complex vector space of dimension >=2 must have nonzero vectors u with <u,u> = 0.  Consequently it's necessary to define hermitian products to define length and angles on complex vector spaces.  Those who wish may redo the exam to turn in Monday for extra credit.  Homework due Friday is:
• p 103: 1
• p 111: 0,2,3,5,6,7,8
• p. 117: 2,3
• 3/11: Hermitian products, Gram - schmidt, existence of Orthogonal bases
• 3/13:  Looked at rank, did a est problem.
• 3/15:  Gave an example of an infinite dimensional vector space V with a positive definite scalar product and a proper subspace W so that the orthogonal compliment of W is 0, thus V is not the direct sum of W and W perp, as is the case in finite dimensions.  Talked some more about the exam and some homework problems.  HW due 3/'22 is:
• p. 122: 1, 2, 6
• p. 125: 2
• p. 131: 2, 3, 4, 6
• p. 134: 3
• p. 138: 1
• 3/18:  Looked at Sylvester's theorem 8.2.
• 3/20:  Talked about the dual space, dual bases, quadratic forms.
• 3/22:  Lecture on what determinents really are.  HW due 4/5
• p. 183 1, 5, 6, 7, 9  Note in problem 6 the limits on the integral should go from minus infinity to infinity, not 0 to 1.
• 4/1:  We talked about the adjoint and transpose of a linear map on a finite dimensional space with scalar or hermitian product.
• 4/3:  Unitary Operators  VII-3
• 4/5  Did some problems.
• 4/8  We talked about Eigenvalues and Eigenvectors VIII-1. HW due 4/12 is:
• p 199: 1,2,3,4,7
  
• p. 212: 1
• 4/10 Characteristic polynomials
• 4/12 We proved the spectral Thm for Normal linear operators. HW due 4/19
• 212: 2, 6, 10, 13
• 218: 3
• 222: 3, 7, 18
• 226: 6, 10
• 4/15:  We looked the consequences of the spectral thm for A real symmetric, that it is orthogonally diagonalizable.  Then we started to look at the consequences for real unitary operators.
• 4/17:  We continued the discussion of normal forms for real unitary operators (e.g. orthogonal matrices). Then we showed that for any linear transformation L from a finite dimensional vector space over C to itself, with a hermitian product, there is an orthonormal basis B so that the matrix of L with respect to B is upper triangular.
• 4/19:  We did a few homework problems.  At the end we looked at the Cayley-Hamilton theorem and quickly sketched the proof.  No homework this week because of the exam Wednesday.
• 4/22:  Proof of C-H thm.  Review.
• 4/24: Exam #2  The exam will cover chapters 5, 7, 8, 9, and 10.  You may bring an 8.5 x 11 sheet of paper with anything you want written on it, both sides.
• 4/26: Went over exam.  HW due 5/3
• 240: 2,3
• 247: 1
• 250: 2
• 253: 1,2,7,8
• 4/29:  Greatest common divisor.  If p and q are poynomials then there are polynomials p' and q' so pp'+qq' = GCD(p,q).
• 5/1:  Application of GCD to linear transformations.    If L:V->V is linear, V fin dim, K=C then V is the direct sum of Ker(L-lambda_i I)^{k_i} where lambda_i are the eigenvalues and k_i is its multiplicity.
• 5/3: Proof of Jordan Canonical form.  I used a more concrete approach than that in the book, but you should read the book's approach also.  HW due 5/10
• 255:15
• 260: 1,2
• 266:1,2,3