Math 416 (Rezk, Section F3H/F4H 2:00 MWF).

Math 416 (Rezk, 2:00 MWF, Section F3H/F4H, 347 Altgeld).

Office hours

See my homepage.

Syllabus

Here is the syllabus.

Textbooks

We will use two books.

Halmos.: Finite-dimensional Vector Spaces, by Paul Halmos, (ISBN-10: 0387900934, ISBN-13: 978-0387900933), currently published by Springer. Here is a table of contents. This book has been around for decades, so inexpensive used copies should not be hard to find.
Hefferon.: Linear Algebra, by Jim Hefferon, is a free textbook, available from the authors webpage . Note that we are still using the 2011 version (which was the latest version as of the start of the course in January.) Here is the pdf file for the 2011 version.

Other resources

Treil.: Linear Algebra Done Wrong, by Sergei Treil, a free textbook with a funny title.

Final exam

Here is the take home part of the exam. It is due Wednesday evening, when you come in for the in class exam.
The in class exam will be in the usual classroom, Wednesday, May 9, 7-10pm. I have made a practice version of the in class exam; here are solutions.

Reading and daily problems

Before February 24.
Friday, February 24.
No DP today.
Read Halmos 32, 33 (Transformations), and Hefferon Three.II (Homomorphisms).
Monday, February 27.
Hand in PS 6.
Wednesday, February 29.
Read Halmos 13,14,15,16 (dual spaces).
Hand in Midterm corrections.
Friday, March 2.
Read Halmos 34 (products).
DP 9: Let \(D,R : \mathcal{P}\to \mathcal{P}\) be the linear operators on real polynomials defined by \( (Df)(t)=f'(t)\) (derivative) and \( (Rf)(t)=f(-t)\). Show that \( DR=-RD \), where \(DR\) and \(RD\) are the composites of the two operators. Explain how this equation implies that \(D\) takes even polynomials to odd polynomials and vice versa.
Monday, March 5.
Read Halmos 35 (inverses), and 36--37 (matrices).
PS 7 due.
Wednesday, March 7.
Matrices, continued. Read Halmos 38 (matrices of transformations); also look at Hefferon Three.III.
DP 10: Halmos 36.7.
Friday, March 9.
Invariance and reducibility. Read Halmos 39, 40.
No DP today
Monday, March 12.
Hand in PS 8.
Change of basis. Read Halmos 46. See also Hefferon Three.V.
Wednesday, March 14.
DP 11: Hefferon Three.V, prob. 1.7 (p. 239).
Friday, March 16.
No DP today.
Rank and nullity. Read Halmos 49, 50.
Spring break.
Monday, March 26.
Reading. Halmos 17 (annihilators).
No DP today.
Wednesday, March 28.
Reading. Hefferon Three.IV.3, Three.IV.4. See also One.III for reduced echelon form.
DP 12: Halmos 17.8, part (a). (numbering fixed.)
Friday, March 30.
Reading. Halmos 54 (Eigenvalues). No Dp.
Monday, April 2.
PS 9 due.
Reading. Halmos 35 (Polynomials).
Wednesday, April 4.
DP 13: Let \(c_0,c_1,c_2\in F\), and let \(A = \begin{bmatrix} 0 & 0 & -c_0 \\ 1 & 0 & -c_1 \\ 0 & 1 & -c_2 \end{bmatrix}\) be a square matrix over \(F\). Show that \(f(A)=0\), where \(f\) is the polynomial defined by \(f(t)=t^3+c_2t^2+c_1t+c_0\). Can this matrix \(A\) be the root of any non-zero polynomial of degree 2 or less?
Minimal polynomials, continued.
Friday, April 6.
Reading: Halmos 30, 31, 53 (alternating forms and determinants).
No DP.
For an amusing proof of the fundamental theorem of algebra using linear algebra, see this exposition by K. Conrad of a proof by Derksen.
Monday, April 9.
Wednesday, April 11.
Second midterm. This will be focused on material we have covered starting from Feb 24, and ending Apr 6: linear maps, representation of linear maps by matrices, dual spaces, range and nullspace, rank and nullity and other kinds of rank, annihilators, eigenvalues and eigenvectors. (Roughly the following sections from Halmos: 13-17, 32-39, 46-47, 49-50, 54. Also, minimal polynomials, which are only covered briefly in a later section of Halmos.) Here is a practice midterm.
Friday, April 13.
Determinant, continued. Read Halmos 30, 31, and 53.
No DP.
Monday, April 16.
Determinant, continued. Read Hefferon Four.I, 1-2.
DP 14. Hefferon (2011 edition) Four.I, problem 1.14 (p. 296).
Wednesday, April 18.
Determinants, continued. Read Hefferon Four.III.
No DP.
Friday, April 20.
PS 11 due.
Monday, April 23.
Direct sums, block decompositions of matrices. Review Halmos 18.
No DP.
Wednesday, April 25.
Generalized eigenspaces.
No DP.
Friday, April 27.
Jordan canonical form. lecture notes on Jordan form
Monday, April 30.
Inner products. lecture notes on inner products and spectral theorem.

Weekly problem sets

PS1. Due Monday, January 23.
1. Hefferon One.I problem 2.23.
2. Hefferon One.I problem 3.15.
3. Hefferon One.I problem 3.17.
4. Consider a (possibly non-homogenous) linear system over R (real numbers) of m equations in n unknowns. Let p₁, p₂, p₃ be n-by-1 column vectors which solve this system. Show that if c₁, c₂, c₃ are real numbers such that c₁+c₂+c₃=1, then the column vector q= c₁p₁+c₂p₂+c₃p₃ is also a solution to the linear sysetm.
PS2. Due Monday, January 30.
1. Hefferon One.I, problem 3.20 (p. 30).
2. Worksheet on fields. (Revised 24 Jan.) g
3. Halmos 4.3.
PS3. Due Monday, February 6.
1. Hefferon Two.I, problem 1.28 (p. 87).
2. Hefferon Two.I, problem 1.29.
3. Halmos 4.5.
4. Halmos 7.1.
5. Halmos 7.4.
6. Hefferon Two.I, problem 2.26 (p. 96).
(That's all.)
PS 4. Due Monday, February 13.
1. Hefferon Two.III, problem 1.24 (page 115) AND Hefferon Two.III, problem 2.19 (page 120).
2. Hefferon Two.III, problem 2.20 (page 120).
3. Halmos 7.8.
4. Halmos 9.1.
5. Let \(F\) be a field, and let \(u=(a,b)\) and \(v=(c,d)\) be two vectors in \(\mathcal{V}=F^2\). Show that \(\{u,v\}\) is a basis of \(\mathcal{V}\) if and only if \(ad-bc \neq 0\).
PS 5. Due Monday, February 20.
1. Halmos 9.3.
2. Halmos 20.3.
3. Let \(\mathcal{U}\) be a vector space, with subspace \(\mathcal{V}\), and assume that \(\mathcal{U}\) is finite dimensional. Show that there exists a subspace \(\mathcal{W}\) of \(\mathcal{U}\) such that \(\mathcal{V}\cap \mathcal{W}=\{0\}\) and \(\mathcal{V}+\mathcal{W}\). (Hint: pick a basis for \(\mathcal{V}\), and extend it to a basis for \(\mathcal{U}\).)
4. Let \(\mathcal{V}\) be a vector space, with basis \(v_1,\dots,v_n\). Given some \(k\in \{1,\dots,n\}\), let \(\mathcal{W}_2= \mathrm{Span}(v_1,\dots,v_k)\) and \(\mathcal{W}_2=\mathrm{Span}(v_{k+1},\dots,v_n)\). Construct an isomorphism between \( \mathcal{W}_1\oplus \mathcal{W}_2 \) and \(\mathcal{V}\).
PS 6. Due Monday, February 27.
1. Hefferon Three.II, problem 1.30. (page 180).
2. Let \(\mathcal{P}\) be the space of real polynomials. Show that each of the following is a linear transformation \( \mathcal{P}\to \mathcal{P}\).
  1. The function \(D : \mathcal{P}\to \mathcal{P}\) defined by \(Df=\frac{df}{dt}\).
  2. For any \(c\in \mathbb{R}\), the function \(S_c: \mathcal{P} \to \mathcal{P}\) defined by \((S_cf)(t) = f(t+c)\).
  3. For any \(q(t)\in \mathcal{P}\), the function \(M_q\colon \mathcal{P}\to \mathcal{P}\) defined by \((M_qf)(t)=q(t)f(t).\)
PS 7. Due Monday, March 5.
1. Halmos 14.2, 14.3, 14.5.
2. Halmos 17.3 (numbering fixed). (Hint for the dimension part of the question: Write \(\mathcal{W}\) for the subspace, and choose a basis \(\langle w_1,\dots,w_k\rangle\) for \(\mathcal{W}\). Let \(v\in \mathcal{V}\) be an element such that \([v,y]\neq 0\) (why does such a thing exist?), and show that \(\langle w_1,\dots,w_k,v\rangle \) is a basis for \(\mathcal{V}\).)
PS 8. Due Monday, March 12.
1. Halmos 38.4.
2. Hefferon Three.III, 1.22, 1.25 (p. 201-202).
3. For each of the following square \(n\times n\) matrices \(A\) with real coefficients, consider the corresponding linear operator \(L:\mathbb{R}^{n\times 1}\to \mathbb{R}^{n\times 1}\) defined by left multiplication by \(A\). In each case, determine all \(L\)-invariant subspaces of \(\mathbb{R}^{n\times 1}\), and determine in each case whether \(L\) is reducible.
  1. \(A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\).
  2. \(A = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\).
  3. \(A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\).
  4. \(A = \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{bmatrix}\).
PS 9. Due Monday, April 2.
1. Let \(T: \mathcal{V}\to \mathcal{W}\) be a linear map. The adjoint of \(T\) is a linear map \(T': \mathcal{W}'\to \mathcal{V}'\) between the dual spaces, defined as follows; if \(f\in \mathcal{W}'\) is a linear functional on \(\mathcal{W}\), then \(T'f\in \mathcal{V}'\) is defined by the formula \[ (T'f)(v) = f(Tv) \] for \( v\in \mathcal{V}\). (See Halmos 44; in bracket notation, the formula becomes \( [v,T'f]=[Tv,f] \), which may be easier to deal with.)
  1. Show that \(T'\) defined above is well-defined (i.e., that \(T'f\) is a linear function), and that \(T'\) is linear.
  2. Show that \(\mathcal{R}(T')\subseteq \mathcal{N}(T)^0\). (Here \(\mathcal{N}(T)^0\) is the annihilator of \(\mathcal{N}(T)\).)
  3. Let \(\mathcal{V}=F^{n\times 1}\) and \(\mathcal{W}=F^{m\times 1}\) be spaces of column vectors, and \(Tx=Ax\) where \(A \in F^{m\times n}\) is a matrix. Show that under the standard identifications \(\mathcal{V}'=F^{1\times n}\) and \(\mathcal{W}'=F^{1\times m}\) of the dual spaces with spaces of row vectors, that the adjoint map \(T': \mathcal{W}'\to \mathcal{V}'\) is given by \(T'y=yA\), where \(y\in F^{1\times m}\).
2. Halmos 51.5. (Hint for part (a): what is the relationship between \(\mathcal{R}(B)\) and \(\mathcal{N}(A)\)? Hint for part (b): choose a basis of \(\mathcal{N}(A)\), extend it to a basis of the whole space, and describe an example of \(B\) in terms of this basis.
3. Hefferon Three.IV (p. 234-5), problems 4.16 and 4.28.
4. A diagonal matrix \(A=(a_{ij})\) is a square matrix such that \(a_{ij}=0\) if \(i\neq j\). Write \(D = \mathrm{diag}(c_1,\dots,c_n)\) for the \(n\times n\) diagonal matrix with \(a_{kk}=c_k\). Determine exactly when such a matrix \(D\) is invertible, and compute its inverse if it exists.
5. Let \(a,b\in F\) be non-zero scalars. Let \(A\) be an \(n\times n\) matrix of the form \(A=P^{-1}DP\) (there was a typo here), where
  1. \(P\) is an invertible \(n\times n\) matrix, and
  2. \(D=\mathrm{diag}(c_1,\dots,c_n)\) is a diagonal \(n\times n\) matrix such that \(\{c_1,\dots,c_n\}\subset\{a,b\}\). (That is, the diagonal entries of \(D\) are either \(a\) or \(b\).
  Show that \(A\) is invertible, and that \(A^{-1}= (a^{-1}+b^{-1})I- (ab)^{-1}A\). (Hint: consider the case when \(A=D\) first.)
PS 10. Due Monday, April 9.
1. Hefferon Five.II, problems 3.34, 3.35 (pp. 363-4). (Numbering in Hefferon is from the 2011 version of book; see above.)
2. Let \(S,T:\mathcal{V}\to \mathcal{V}\) be linear operators which commute; that is, \(TS=ST\). Show that every eigenspace \(\mathcal{E}_T(\lambda)\) of \(T\) is an invariant subspace for \(S\).
  Use this to show that if \(v_1,\dots,v_n\) is a basis of eigenvectors for \(T\) with distinct eigenvalues, then it is also a basis of eigenvectors for any operator \(S\) which commutes with \(T\).
3. Show that if \(S,T:\mathcal{V}\to \mathcal{V}\) are linear operators, and \(c\in F\) is a scalar, such that \(TS=cST\), then \(S\) maps elements of \(\mathcal{E}_T(\lambda)\) into \(\mathcal{E}_T(c\lambda)\).
4. Show that if \(S,T:\mathcal{V}\to \mathcal{V}\) are linear operators, and \(c\in F\) is a scalar, such that \(TS=ST+cS\), then \(S\) maps elements of \(\mathcal{E}_T(\lambda)\) into \(\mathcal{E}_T(\lambda+c)\).
5. This exercise asks you to verify some examples of the phenomena of the previous exercises. Let \(\mathcal{C}^\infty\) be the space of smooth real valued functions on \(\mathbb{R}\),
  1. Let \(c\in \mathbb{R}\), and define a shift operator \(S_c: \mathcal{C}^\infty\to \mathcal{C}^\infty\) by \[ (S_cf)(t) = f(t-c). \] (The function \(S_cf\) is the one whose graph is the graph of \(f\) shifted to the right by \(c\) units.)
    Show (i) that \(D=d/dt\) commutes with \(S_c\), and (ii) verify that the function \(e^{at}\) is an eigenvector for both \(D\) and \(S_c\).
    Show that (iii) \(D^2\) commutes with \(S_c\), and that (iv) the subspace \(\mathcal{W}_a = \mathrm{Span}(\cos at,\sin at )\) is invariant under both \(D^2\) and \(S_c\). (v) Must \(\mathcal{W}_a\) contain eigenvectors for \(S_c\)?
  2. Let \(c\in \mathbb{R}\), and define a dilation operator \(F_c:\mathcal{C}^\infty\to \mathcal{C}^\infty\) by \[ (D_cf)(t) = f(ct). \]
    Show (i) that \(DF_c=cF_cD\), and that (ii) \(F_c\) carries \(\mathcal{V}_a=\mathrm{Span}(e^{at})\) into \(\mathcal{V}_{ca}\).
  3. Let \(c\in \mathbb{R}\), and define an operator \(T_c\colon \mathcal{C}^\infty\to \mathcal{C}^\infty\) by \[ (T_cf)(t) = e^{ct}f(t). \]
    Show (i) that \(DT_c=T_cD+cT_c\) (typo fixed), and verify directly that (ii) \(T_c\) carries the subspace \(\mathcal{V}_a=\mathrm{Span}(e^{at})\) into \(\mathcal{V}_{a+c}\).
PS 11. Due Friday, April 20.
1. Hefferon Four.I, problems 1.1, 1.3 (p.295).
2. Hefferon Four.I, problem 2.7, 2.13. (p. 300)
3. Hefferon Four.III, problems 1.14, 1.20 (p. 328)
PS 12. Due Monday, April 30.
1. Halmos 53.9 (fixed) (p. 102). (In this problem, assume that the matrices \(A,B,C,D\) are square and of the same size.)
2. Halmos 55.1 (p. 106).
3. Do the following worksheet on real and complex eigenvalues.

Contact info.

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office: Illini Hall 242
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Last modified 7 May 2012 by Charles Rezk.

Department of Mathematics
College of Liberal Arts and Sciences
University of Illinois at Urbana-Champaign
273 Altgeld Hall, MC-382
1409 W. Green Street, Urbana, IL 61801 USA
Department Main Office Telephone: (217) 333-3350 Fax (217) 333-9576