Question

**Choose either true or false for each**
**statement**

a. There is a vector [b1 b2] so that the set of solutions to

1 | 0 | 1 |

0 | 1 | 0 |

[ x1, x2 , x3,] =[b1b2] is the z-axis.

b. The homogeneous system Ax=0 has the trivial solution if and only if the system has at least one free variable.

c. If x is a nontrivial solution of Ax=0, then every entry of x is nonzero.

d. The equation Ax=b is homogeneous if the zero vector is a solution.

e. A homogeneous system is always consistent

f. The solution set of a consistent inhomogeneous system Ax=b is
obtained by translating the solution set of Ax=0.

Answer #1

a. FALSE. The solution to the equation A (x_{1},
x_{2} , x_{3})^{T}
=(b_{1},b_{2})^{T} is (x_{1},
x_{2} , x_{3})^{T}
=(b_{1}-x_{3}, b_{2},
x_{3})^{T} . Whatever be the value of
b_{1},b_{2}, no solution of the equation can be the
vector (0,0,x_{3})^{T} i.e. the z-axis or, the
x_{3} -axis.

b. FALSE. The homogeneous system Ax = 0 always has the trivial solution.

c. FALSE. If x is not equal to the zero vector, and Ax = 0, then x is a non-trivial solution.

d. TRUE. If 0 is a solution, then b = A0 = 0 for any matrix A.

e. TRUE. It always has the trivial solution.

f. TRUE.

For each statement below, either show that the statement is true
or give an example showing that it is false. Assume throughout that
A and B are square matrices, unless otherwise specified.
(a) If AB = 0 and A ̸= 0, then B = 0.
(b) If x is a vector of unknowns, b is a constant column vector,
and Ax = b has no solution, then Ax = 0 has no solution.
(c) If x is a vector of...

A system of linear equations is said to be homogeneous if the
constants on the right-hand side are all zero. The system
2x1 − x2 + x3 + x4 = 0
5x1 + 2x2 − x3 − x4 = 0
−x1 + 3x2 + 2x3 + x4 = 0
is an example of a homogeneous system. Homogeneous systems
always have at least one solution, namely the tuple consisting of
all zeros: (0, 0, . . . , 0). This solution...

Find the condition on b1, b2 and b3 for the following system of
equations to
have a solution.
x1 + 3x2 + 2x3 + 10x4 = b1
2x1 + 3x2 + 5x3 + 3x4 = b2
5x1 + 9x2 + 12x3 + 16x4 = b3
(b) Find the complete solution for (b1, b2, b3) = (0, 1, 2).

x1-5x2+x3+3x4=1
2x1-x2-3x3-x4=3
-3x1-3x3+7x3+5x4=k
1 ) There is exactly one real number k for which the system has
at least one solution; determine this k and describe all solutions
to the resulting system.
2 ) Do the solutions you found in the previous part form a
linear subspace of R4?
3 ) Recall that a least squares solution to the system of equations
Ax = b is a vector x minimizing the length |Ax=b| suppose that
{x1,x2,x3,x4} = {2,1,1,1}
is a...

For parts a and b, find a basis for the solution set of the
homogeneous linear systems. Show all algebraic steps.
a. x1 + x2 + x3 = 0.
x1 - x2 - x3 = 0
b. x1 + 2x2 - 2x3 + x4 = 0.
x1 - 2x2 + 2x3 + x4 = 0.
for parts c and d use your solutions to parts a and b to find
all solutions to the following linear systems. show all algebraic...

Is each statement true or false? If true, explain why; if false,
give a counterexample.
a) A linear system with 5 equations and 4 unknowns is always
inconsistent.
b) If the coefficient matrix of a homogeneous system has a
column of zeroes, then the system has infinitely many solutions.
(Note: a homogeneous system has augmented matrix [A | b] where b =
0.)
c) If the RREF of a homogeneous system has a row of zeroes, then
the system has...

True/ false
a- If the last row in an REF of an augmented matrix is [0 0 0 4
0], then the associated linear system is inconsistent.
b-The equation Ax=b is consistent if the augmented matrix [A b]
has a pivot position in every row.
c-The set Span{v} for a nonzero v is always a line that may or
may not pass through the origin.

1. If x1(t) and x2(t) are solutions to the differential
equation
x" + bx' + cx = 0
is x = x1 + x2 + c for a constant c always a solution? Is the
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Show the works
2. Write sown a homogeneous second-order linear differential
equation where the system displays a decaying oscillation.

1.) either solve the given system of equations, or else show
that there is no solution.
x1 + 2x2 - x3 = 2
2x1 + x2 + x3 = 1
x1 - x2 + 2x3 = -1
2.) determine whether the members of the given set of vectors
are linearly independent. If they are linearly dependent, find a
linear relation among them.
(a.) x(1) = (1, 1, 0) , x(2) = (0, 1, 1) ,
x(3) = (1, 0, 1)...

Find the general solution to the following linear system around
the equilibrium point:
x '1 = 2x1 + x2
x '2 = −x1 + x2
(b) If the initial conditions are x1(0) = 1 and x2(0) = 1, find
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(c) Plot the exact vector field (precise amplitude of the
vector) for at least 4 points around the equilibrium, including the
initial condition.
(d) Plot the solution curve starting from the initial condition
x1(0)...

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