Question

T: R^3 ----> R^5 such that T(x), then...

a. A^-1 is 3x 5 matrix

b. the mapping cannot be onto

c. the set of solutions to Ax=0 is infinite

d. A has at least 2 free variables

e. none of the above

Answer #1

1. Define T : R 2 → R 2 by T(x, y) = (3x + 2y, 5x + y).
(a) Represent T as a matrix with respect to the standard basis
for R 2 .
(b) First, show that B = {(1, 1),(−2, 5)} is another basis for R
2 . Then, represent T as a matrix with respect to B.
(c) Using either (a) or (b), find the kernel of T.
(d) Is T an isomorphism? Justify your answer....

1) Find all a in R such that x^3 + ax^2 + 3x + 15 is strictly
increasing near x = 1.
2) Find all a in R such that ax^2 + 3x + 5 is strictly
increasing on the interval (1,2).

n x n matrix A, where n >= 3. Select 3 statements from the
invertible matrix theorem below and show that all 3 statements are
true or false. Make sure to clearly explain and justify your
work.
A=
-1 , 7, 9
7 , 7, 10
-3, -6, -4
The equation A has only the trivial solution.
5. The columns of A form a linearly independent set.
6. The linear transformation x → Ax is one-to-one.
7. The equation Ax...

Let A = [1 5 ; 3 1 ; 2 -4] and b = [1 ; 0 ; 3] (where semicolons
represent a new row)
Is equation Ax=b consistent?
Let b(hat) be the orthogonal projection of b onto Col(A). Find
b(hat).
Let x(hat) the least square solution of Ax=b. Use the formula
x(hat) = (A^(T)A)^(−1) A^(T)b to compute x(hat). (A^(T) is A
transpose)
Verify that x(hat) is the solution of Ax=b(hat).

The augmented matrix represents a system of linear equations in
the variables x and y.
[1 0 5. ]
[0 1 0 ]
(a) How many solutions does the system have: one, none, or
infinitely many?
(b) If there is exactly one solution to the system, then give
the solution. If there is no solution, explain why. If there are an
infinite number of solutions, give two solutions to the system.

Problem 2. (20 pts.) show that T is a linear transformation by
finding a matrix that implements the mapping. Note that x1, x2, ...
are not vectors but are entries in vectors. (a) T(x1, x2, x3, x4) =
(0, x1 + x2, x2 + x3, x3 + x4) (b) T(x1, x2, x3, x4) = 2x1 + 3x3 −
4x4 (T : R 4 → R)
Problem 3. (20 pts.) Which of the following statements are true
about the transformation matrix...

If a, b ∈ R with a not equal to 0, show that the infinite set
{1,(ax + b),(ax + b)2 ,(ax + b)3 , · · · } of
polynomials is a basis for F[x].

Let A be a given (3 × 3) matrix, and consider the equation Ax =
c, with c = [1 0 − 1 ]T . Suppose that the two vectors
x1 =[ 1 2 3]T and x2 =[ 3 2 1] T are
solutions to the above equation.
(a) Find a vector v in N (A).
(b) Using the result in part (a), find another solution to the
equation Ax = c.
(c) With the given information, what are the...

1) Solve by Substitution.
{x+2y=−1
{-6y=3+3x
a) One solution:
b) No solution
c) Infinite number of solutions
2) Solve by Substitution
{4x+y=−15
{2y=-27-8x
a) One solution:
b) No solution
c) Infinite number of solutions
3) Solve the system by elimination.
{−x−2y=4
{2x+2y=-8
a) One solution:
b) No solution
c) Infinite number of solutions

q.1.(a)
The following system of linear equations has an infinite number
of solutions
x+y−25 z=3
x−5 y+165 z=0
4 x−14 y+465 z=3
Solve the system and find the solution in the form
x(vector)=ta(vector)+b(vector)→, where t is a free
parameter.
When you write your solution below, however, only write the part
a(vector=⎡⎣⎢ax ay az⎤⎦⎥ as a unit column vector with the
first coordinate positive. Write the results accurate to
the 3rd decimal place.
ax =
ay =
az =

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