LINEAR ALGEBRA and
DIFFERENTIAL EQUATIONS
Using MATLAB

Martin Golubitsky and Michael Dellnitz

Corrections

Page	Line	Correction
2		In paragraph 2 interchange $m$ with $n$ and $m'$ with $n'$
3		Interchange problems 2 and 3
62	7	and the $m$ vector
62	8	$b = \left(\begin{array}{c} b_1 \vdots b_m \end{array}\right)$
63	-3	vector using matrix product in Matlab.
66	(3.1.7)	Entry of $A$ in row 4, column 5 should be 8.0
75	-6	...what functions $L:{\bf R}\to{\bf R}$ are linear.
75	-5	Since we are looking at the special case of linear mappings on ${\bf R}$, we note that $x$ is a
		real number as well as a vector. Thus
75	-3,-2	In addition, if $L:{\bf R}\to{\bf R}$ is linear, we can use Definition 3.3.1(b) ...
103	5	$L_A(P)=\{Az: z\in P\}$
103	-9	$\vert L_A(P)\vert = \cdots$
103	-5	$P = L_B(S)$.
103	-4	$\vert L_A(P)\vert = \vert L_{AB}(S)\vert$
105	3	$L_A(S)$
138	-5	$g(x) \neq 0$
146	16	Sinks: $a < 0$ and $d<0$
152	4	$(x(t),y(t))=(\cos t, -\sin t)$ is a solution
174	8	$-3.0253 e^{2.3543t}\left(\begin{array}{r} -0.9830 -0.1838 \end{array}\right) + \cdots$
188	-4	$0v = \mathbf{0}$
189	-6	Let $W$ be a nonempty subset ...
199		Change $A$ to $W$ in lines 1, 3, 7, 10, 12
200		Change $A$ to $W$ in lines 7, 10, 16, 18
203		Change $A$ to $W$ in lines 5, 8, 10, 11, 15, 16
204		Change $A$ to $W$ in lines 6, 8, 11
207	8	equations
207	11	on the matrix $M$ defined in Lemma 5.5.4.
228	16	eigenvalue of the real $2\times 2$ matrix $C$
237	10	$C^4=I_2$
238	1	To compute the matrix exponential, MATLAB ...
241	-8	$\dot{X}=\left(\begin{array}{cc} \lambda_1 & 1 0 & \lambda_1\end{array} \right)X$
248	-4	by $v_2$, we calculate
252	10	$e^{tC} = \frac{1}{\lambda_1-\lambda_2} \cdots$
285	-4	$n\ge 3$
289	3, 5, 7	Change $I_n$ to $I_m$
294	-9	induction. The simplest meaningful case ($n=2$; $k=1$) is easily verified.
298	8, 9	Replace ``Note here that'' with ``By definition''
298	11	and therefore, by Theorem 3.7.8, there exists
323	5	...Lemma 9.1.3,
323	14	Theorem 9.1.2 states ...
333	15	...$z_2 =(-1,2)$
333	-3	$C_{\mathcal{WZ}} = \left(\begin{array}{rr} 1 & 1 1 &-2\end{array}\right)^{-1}\left(\begin{array}{rr} 1 & -1 3 &2\end{array}\right)$
	-2	$= \frac{1}{3}\left(\begin{array}{rr} 2 & 1 1 &-1\end{array}\right)\left(\begin{array}{rr} 1 & -1 3 &2\end{array}\right)$
	-1	$= \frac{1}{3}\left(\begin{array}{rr} 5 & 0 -2 &-3\end{array}\right)$
350	10	made in using ...
350	-3	b0(1) = -3.8197
350	-1	b0(3) = 0.0443
365	-1	replace $w_i^2$ with $w_j^i$
376	9	using the notion of differentiability in two variable calculus, but ...
387	-4	$x^2+y^2=r_0^2$
401	4 & 7	$\dot{y} = y(-4+x)$
408	-1	...that the linearization at the origin ...
409	-1	In Figure 12.9 -- no change in weight of dots
412	5	Delete the sentence ``We discuss homoclinic ...Section 12.2.''
413	1	In Figure 12.12 -- no change in weight of dots
414	1	In Figure 12.13 -- branch of limit cycles off top branch should be dotted
423	1	In Figure 12.16 -- branch of limit cycles off top branch should be dot-dashed
427		problem 3. $\rho=-0.04$
428	6	(11.4.5)
429	6	$f(x,\rho) = \frac{1}{2}ax^2 + \rho$
449	6	if there is an invertible complex
454	1	e13_2_13
454	-7	in real block diagonal form
466	-7	$\begin{array}{lrr} {\rm null2 =} & \\ & 0.2690 & 0.1605 \\ & 0.6773 & -0.6937 \\ & -0.6712 & -0.6749 \\ & -0.1357& 0.1935 \end{array}$
467	1	$\begin{array}{lr} {\rm v11 =} & \\ & -0.8093 \\ & 2.8327 \\ & 3.2374 \\ & -0.8093 \end{array}$
467	6	Note that V11 is nonzero and a multiple of the eigenvector. Thus, we ...
482		Equation (14.1.2): $+\frac{1}{(k-1)!}t^{k-1}N^{k-1}$
486	4,-14 - -12	Change $I_n$ to $I_4$
490	-7	$s\times s$ matrix
490	-5	$u\times u$ matrix
490	-3	$c\times c$ matrix
491	12	$z(t)$ can be either bounded or grow.
493	5	e14_2_4
494		In problem 14.2.2 change to $+z^3$ in $\dot{x}$ equation and $-x^3$ in $\dot{z}$ equation
507	11	$[t,x] = \mbox{ode45}('f14\_4\_4', [0 50], [0.5, 0.4, 0.3]');$
539	-2	$\{t^je^{\lambda_\ell t}: \cdots$
541	-6	with algebraic multiplicity
566	7	$x_p(t) = -e^{2t}$
566	10	$x(t) = -e^{2t} + \alpha e^{3t}$
566	11	$\alpha = 2$
599	-1	$c(t) = \int_{t_0}^t g(\tau)e^{-H(\tau)}d\tau + x_0e^{-H(t_0)}$
602	-5	$X_j(0) = v_j$.
620	-1	$\frac{v^2}{2} = 2(\ln(t)-\ln(2)) + \frac{9}{2}$
621	2	$v(t) = \sqrt{9+4\ln\frac{t}{2}}$
621	4	$x(t)=tv(t) = t\sqrt{9+4\ln\frac{t}{2}}$
635	9	Change $V(x)$ to $V'(x)$ in (17.7.5)
635	-3, -2	Change $V(x)$ to $V'(x)$
636	3,6	Change $V(x)$ to $V'(x)$
637	6, 7	Change $V(x)$ to $V'(x)$
675	5	Section 1.1, Problem 3: answer given is to Problem 2. Problem 3 answer is: $(8,4,12)$
681	-4	3. $X(t) =\frac{7}{3}e^t \cdots$
684	6	$\left(\begin{array}{r} x(t) y(t) \end{array} \right) = \left(\begin{array}{... ...{2t}(\cos(3t) - 2\sin(3t)) -e^{2t}(\sin(3t) + 2\cos(3t)) \end{array} \right)$
695	-4	7. $\frac{1}{2e^{2-t}-1}$
Corrections to Solutions Manual
8	-2	y = 2 - x.*sin(x.^2 - 1);

The toolbox file e12_2_14.pps is incorrect. The correct file can be downloaded from

ftp.math.uh.edu/pub/laode/matlab6_files/e12_2_14.pps