Question

Suppose you have an n×n homogenous linear system of equations.
Make a statement classifying exactly when you have a unique
solution (what is it?) if the system is written in RREF. Prove
(i.e. explain why) your statement is true.

Answer #1

Geometrically, why does a homogenous system of two linear
equations in three variables have infinitely many solutions? If the
system were nonhomogeneous, how many solutions might there be?
Explain this geometrically.

Is it possible for a system of linear equations with more
equations than unknowns to have a unique solution? What about fewer
equations than unknowns? If your answer to either question is
“yes,” provide an example.

Let A be a n × n matrix, and let the system of linear equations
A~x = ~b have infinitely many solutions. Can we use Cramer’s rule
to find x1? If yes, explain how to find it. If no, explain why
not.

Is the following statement true or false: If a system of two linear differential equations is saddle point stable, the variables can move against their steady state values over time. Explain your answer.

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.

4. Suppose that we have a linear system given in matrix form as
Ax = b, where A is an m×n matrix, b is an m×1 column vector, and x
is an n×1 column vector. Suppose also that the n × 1 vector u is a
solution to this linear system. Answer parts a. and b. below.
a. Suppose that the n × 1 vector h is a solution to the
homogeneous linear system Ax=0.
Showthenthatthevectory=u+hisasolutiontoAx=b.
b. Now, suppose that...

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...

A linear system of equations Ax=b is known, where A is a matrix
of m by n size, and the column vectors of A are linearly
independent of each other. Please answer the following questions
based on this assumption, please explain it, thank you~.
(1) To give an example, Ax=b is the only solution.
(2) According to the previous question, what kind of inference
can be made to the size of A at this time? (What is the size of...

Prove that for a square n ×n matrix A, Ax = b (1) has one and
only one solution if and only if A is invertible; i.e., that there
exists a matrix n ×n matrix B such that AB = I = B A.
NOTE 01: The statement or theorem is of the form P iff Q, where
P is the statement “Equation (1) has a unique solution” and Q is
the statement “The matrix A is invertible”. This means...

For each part below, give an example of a linear system of
three equations in three variables that has the
given property. in each case, explain how you got your answer,
possibly using sketches.
(a) has no solutions
(b) has exactly one solution which is (1, 2, 3).
(c) any point of the line given parametrically be (x, y, z) = (s
− 2, 1 + 2s, s) is a solution and nothing else is.
(d) any point of the...

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