Solutions to Section 1.1
1. FALSE. A derivative must involve some derivative of the function y = f (x), not necessarily the rst
2. TRUE. The initial conditions accompanying a dierential equation consist of the values of y,
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Homework #2 Solutions
1. Consider the differential equation
(3.) Find the general solution to this differential equation.
We may View this as a rst order linear ODE. We put this in standard form, yielding
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Homework #6 Solutions
1. Determine whether the following sets of vectors are linearly independent or not.
1 x + x2 , 2 2x + x2 in P2 .
Its fairly easy to see that neither polynomial is a multiple of the other. (For instance, in
order to get t
(4) Find the general solution of each of the following systems of linear equations:
As a check on your arithmetic, note that the matrix of the system in part (a) has rank
2 and that the matrix of the system in part (b) has rank 3. (10 points for Part (a)
Homework #5 Solutions
1. Determine whether the following subsets of C 2 (R) are vector spaces or not (that
is, whether theyre subspaces of C 2 (R) or not).
(a) The set of functions f satisfying f (1) = f (2) 1.
This not a subspace. One quick of w
Homework #8 Solutions
1. For each of the following the linear transformations, nd a basis for the kernel and
(a) T : P2 R3 , T a + bx + cx2 = (a + b + c, b c, a + 2b) .
The kernel will be given by all a, b, and c such that
a + b + c =
Tuesday, Sep. 6 (Dr. Colls sections)
Monday, Sep. 5 (all other sections)
1.1 Differential Equations Everywhere - 1.6 First-Order Linear Differential Equations
Note: This homework is worth 10 points. Each pro
Tuesday, Sep. 20 (Dr. Colls sections)
Monday, Sep. 19 (all other sections)
2.4 Row-Echelon Matrices and Elementary Row Operations - 2.6 The Inverse of a Square Matrix
Note: This homework is worth 10 points.
Exercises for Section 4.7
In exercises 1-5, V = R2 , E =
is the standard basis of V , and B =
are bases of V .
, nd [u1 ]B .
1. (a) If u1 =
(b) If [u2 ]B =
, nd u2 .
, nd [u3 ]C .
2. (a) If u3 =
, nd u4 .
Tuesday, Sep. 13 (Dr. Colls sections)
Monday, Sep. 12 (all other sections)
1.6 First-Order Linear Differential Equations - 2.3 Terminology for Systems of Linear Equations
Note: This homework is worth 10 poin