(8) TRUE/FALSE: Circle either T or F. No justification is needed.

(a) (T : F) Each line in R n is a one-dimensional subspace of R n .

(b) (T : F) The determinant of A is the product of the pivots in any echelon form U of A, multiplied by (−1)r , where r is the number of row interchanges made during row reduction from A to U.

(c) (T : F) Adding a multiple of one row to another does not affect the determinant of a matrix.

(d) (T : F) det(A + B) = det(A) + det(B).

(e) (T : F) If the columns of A are linearly dependent, then det A = 0.

(f) (T : F) det AT = (−1) det A.

(g) (T : F) The determinant of A is the product of the diagonal entries in A.

(h) (T : F) If det A is zero, then two rows or two columns are the same, or a row or a column is zero.

(i) (T : F) If two row interchanges are made in succession, then the determinant of the new matrix is equal to the determinant of the original matrix.

Let H be the subset of all skew-symmetric matrices in M_3x3

a.) prove that H is a subspace of M_3x3 by checking all three conditions in the definition of subspace.

b.) Find a basis for H. Prove that your basis is actually a basis for H by showing it is both linearly independent and spans H.

c.) what is the dim(H)

Show that for all σ ∈ Sn we have sgn(σ) = sgn(σ−1). Does σ = (1,2,3,5,4)−1 ∈ S13 belong to the alternating group A13? Justify your answer

Use the Cauchy Criterion to prove the Bolzano–Weierstrass Theorem, and find the point in the argument where the Archimedean Property is implictly required. This establishes the final link in the equivalence of the five characterizations of completeness discussed at the end of Section 2.6.

Recall that a standard 52-card deck has four suits, ♥, ♦, ♣, and ♠, each of which has13cards, one each of the following kinds, A, K, Q, J,10,9,8,7,6,5,4,3,and2. A hand of seven (7) cards is drawn at random from such a deck. (This means that you get the cards as a group, in no particular order, and with no possible way of getting the same card twice in the hand.) Find the probability that the hand . . .

1. 2. 3.

4. 5.

... is a flush, i.e. all the cards in the hand are from the same suit. [1]

. . . has four cards of the same kind. [1]

. . . has exactly three cards of one kind, two cards of another kind, and two cards of yet another kind. [1]

. . . has cards of seven different kinds. [1]

... is a straight, i.e. a set of cards that can be arranged to be consecutive with no gaps in the sequence AKQJ1098765432, where we allow the sequence to wrap around the end. (So 3 2 A K Q J 10 would count as a straight, for example.) [1]

Show all your work!

Find the infinite series solution about x = 0 for the following DE, using Bessel's, Legrende's, or Frobenius method equations.

3x^2y" + 2xy' + x^2y = 0

Find the first 4 non-zero terms in the series expansion. Do not use k=n substitutions

How do you imagine the graph of a three variable system of three equations would look? How about the graph of a system of four equations in four variables?
First, think about a two variable system, the corresponding graph and what each equation in the system represents. Then, expand this idea to three variables and discuss the possible cases for solutions to the system. Finally, extend the idea to four variables.
If you produce three items that each require three inputs, then you can use a system of three equations in three variables to solve. As an example, you could make small , medium and large pizzas ( these would be your variables). Your equations ( constraints) would come from limitations on how much dough , pizza sauce and cheese you have.
How can you relate a point, a line segment, a square and a cube?
How can you relate a point, a line segment , a circle and a sphere? Hint, start with the smallest of these and think about how you could build up to the next one and then the next one.

It is college allgebra.

Let A be an n × n matrix which is not 0 but A² = 0. Let I be the identity matrix.

a)Show that A is not diagonalizable.

b)Show that A is not invertible.

c)Show that I-A is invertible and find its inverse.

Use ‘Reduction of Order’ to find a second solution y2 to the given ODEs:

(a) y′′+2y′+y=0, y1 =xe−x

(b) y′′+9y=0, y1 =sin3x
(c) x2y′′+2xy′−6y=0, y1 =x2

(d) xy′′ +y′ =0, y1 =lnx

Solve a system of equations:
1-
2x = 5 mod 15  
3x = 1 mod 4

2-
x = 5 mod 15
x = 2 mod 12

(Hint: Note that 15 and 12 are not relatively prime. Use the Chinese remainder
theorem to split the last equation into equations modulo 4 and modulo 3)

Let W be the set of P₄ consisting if all polynomials satisfying the conditions p(-2)=0.

a.) prove that W is a subspace of P₄ by checking all 3 conditions in the definition of subspace.

b.) Find a basis for W. Prove that your basis is actually a basis for W by showing it is both linearly independent and spans W

c.) what is the dim(W)

For the wheel W₉ find the minimal coloring. Show that the coloring is sufficently large by enumnerating it. Explain why it may not be colored with fewer.

Problem 7.3. Let f (x, y) = x6 + 3xy + y2 + y4.
(a) Show that f remains unchanged if you replace x by −x and y by −y. Hence,
if (x, y) is a critical point of f, so is (−x, −y). Thus, critical points other than
(0, 0) come in ± pairs.
140 7 Optimization in Several Variables
(b) Compute the partial derivatives of f . Show that solve applied directly to
the system fx = fy = 0 fails to locate any of the critical points except for (0, 0).
(c) Let’s compensate by eliminating one of the variables and then using solve
followed by double. First solve for y in terms of x in the equation fx = 0.
Substitute back into the formula for fy and then apply first solve and then
double. You should end up with three critical values of x, giving a total of
three critical points. Find the numerical values of their coordinates. (Be sure
you have set x and y to be real; otherwise you will also end up with many
irrelevant complex critical points.)
(d) Confirm the calculation of the critical points by graphing the equations fx =
0 and fy = 0 on the same set of axes (using fimplicit and hold on). You
should see exactly one additional pair of critical points (in the sense of (a)).
(e) Classify the three critical points using the second derivative test.
(f) Apply fminsearch to f with the starting values (1, 1) and (0, 0). Show
that in the first case you go to a minimum and that in the second case you stay
near the saddle point.

Problem 7.1. Let f (x, y) = x4 − 3xy + 2y2.
(a) Compute the partial derivatives of f as well as its discriminant. Then use
solve to find the critical points and to classify each one as a local maximum,
local minimum, or saddle point.
(b) Check your answer to (a) by showing that fminsearch correctly locates
the same local minima when you start at (0.5, 0.5) or at (−0.5, 0.5).
(c) What happens when you apply fminsearch with a starting value of
(0, 0)? Explain your answer.
(d) What are the values of f at the extrema? Now, using fmesh, graph the function
on a rectangle that includes all the critical points. Experiment with view
and axis until you get a picture that shows the behavior near the critical points.
Use the graph and all the previous data to justify the assertion: Sometimes symbolic
and/or numerical computations are more revealing than graphical information.

What is an equilibrium solution? I am in an Elementary Differential Equations course. The problem is dy/dx = e^y-1; (0,0) and (1,1) and wants the equilibrium solutions. Been a few years since I've taken a math course I'm not finding any good explainations for what it is, thanks.