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MATH 141: Linear Analysis I Homework 05 solved

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1. For pictures labeled ”a” and ”d” inn the exercise titled ”Intersection Of Three Planes” done in lecture,
write down a 3×3 system (that is, a system of 3 equations with 3 unknowns) that corresponds to that
picture. Explain why you think your system is valid. In other words, use sentences, math notations and
calculations, etc. (whatever is necessary), to show that the picture representing the system you wrote down
indeed looks like the given picture.
2. Complete the following sentences about linear systems of equations by either choosing the correct words
or filling in the blanks. Answer the question in part (c) too.
(a) (Strang §1.2 #14) For two linear equations in three unknowns x, y, z, the row picture will show (2
or 3) (lines or planes) in (two or three)-dimensional space. The column picture is in (two or three)-
dimensional space. The solutions normally lie on a .
(b) (Strang §1.2 #15) For four linear equations in two unkwons x and y, the row picture shows four
in -dimensional space. The column picture is in -dimensional space. The equations have no
solution unless the vector on the right-hand side is a combination of .
(c) (Strang §1.2 #20 adapted) Normally 4 ”hyperplanes” in four-dimensional space meet at a . In
order to determine what linear combination of




1
0
0
0




,




1
1
0
0




,




1
1
1
0




,




1
1
1
1




produces ~b =




3
3
3
2




, which system
of equations for x, y, z, t do you need to solve? Solve that system.
3. (based on Strang §1.3 #9) Consider the following system of equations.
3×1−2×2=b1
6×1−4×2=b2
You will sketch two different graphs. You need relatively accurate graphs in order to answer part (f) which
is the main point of this whole exercise.
(a) If we write the above system using matrix notations A~x = ~b, what is the coefficient matrix A, what is
~x, and what is ~b?
(b) What is the definition of the nullspace of a given matrix? Describe the nullspace N(A) of this particular A as a span.
(c) What is the definition of the column space of a given matrix? Describe the column space C(A) of this
particular A by describing the condition(s) that the components of each vector in C(A) have to satisfy.
(d) Choose three concrete right-hand vectors ~b1,
~b2,
~b3 for which the system is inconsistent. On your first
graph, sketch N(A), C(A), and~b1,
~b2,
~b3.
(e) Choose two concrete non-zero right-hand vectors ~b4,
~b5 for which the system is consistent. Find all
solutions to the two systems for with ~b4 and ~b5 respectively. How many solutions are there for each
system? On your second graph, sketch N(A), C(A),
~b4,
~b5, and all solutions corresponding to ~b4 and
~b5.
(f) Study your two graphs and notice any special relationship(s) among various objects. Complete the
following sentences.
i. In order for A~x = ~b to have a solution, the right-hand vectors ~b .
ii. In the case that A~x = ~b is solvable, all its solutions form .
MATH 141: Linear Analysis I Homework 05 Fall 2019
4. (cf. Strang §1.3 #18) Suppose that ~x0 and ~x1 are two different solutions to the same linear system of equations A~x = ~b, where A is the coefficient matrix, ~x the unknown, and~b may or may not be the zero vector.
(a) Write down, in terms of ~x0 and ~x1, a solution to A~x = ~0. How many other solutions to A~x = ~0 can
you also write down?
(b) Write down, in terms of ~x0 and ~x1, a third solution to A~x = ~b. How many other solutions to A~x = ~b
can you also write down?
(c) (my own) Explain how this problem is related to the second sentence you need to complete in part (f)
of problem 3.
5. A student is trying to figure out for what right-hand vector~b =


b1
b2
b3

 a linear system of equations A~x = ~b is
solvable. After she performed Gaussian elimination, the result, not the original augmented matrix, looks
like





1 2 3

b1
0 0 3

b2 − b1
0 0 0

b3 − 2b1





(a) What is the rank of the matrix A? Which columns are pivot columns? Which variables are free
variables?
(b) What condition(s) do b1, b2, b3 have to satisfy to make the system solvable?
(c) According to the first sentence you need to complete in part (f) of problem 3, by describing what conditions b1, b2, b3 have to satisfy for the system to be solvable, we are actually describing
of the coefficient matrix A.
(d) Construct a concrete ~b satisfying your conditions in part (b). Solve the system.
(e) Find N(A).
(f) Note that we do not need to know what A is in order to answer all previous questions. However, we
can find A. Find A.
6. (a) Does the Gaussian elimination process change the nullspace of a matrix? In order words, if we perform Gaussian elimination on a matrix A to get its upper echelon form U, are N(A) and N(U) different? Why or why not?
(b) Repeat the discussion above for column space.
7. (a) (Strang §2.2 #35) What conditions on b1, b2, b3, b4 make each system solvable? Solve for ~x.




1 2
2 4
2 5
3 9





x1
x2

=




b1
b2
b3
b4








1 2 3
2 4 6
2 5 7
3 9 12






x1
x2
x3

 =




b1
b2
b3
b4




(b) By describe what conditions b1, b2, b3, b4 have to satisfy so that the system is solvable, we are actually
describing of the coefficient matrix. Explain why.
8. (postponee to next week) (Strang §2.2 #39) Explain why all these statements are all false (all statements are
about solving linear systems A~x = ~b):
(a) The complete solution is any linear combination of ~xparticular and ~xnullspace.
MATH 141: Linear Analysis I Homework 05
(b) A system A~x = ~b has at most one particular solution.
(c) The solution ~xparticular with all free variables zero is the shortest solution (minimum length k~xk). (Find
a 2×2 counterexample.)
(d) If A is invertible there is no solution ~xnullspace in the nullspace. (Lei Yue’s comment: you do not even
need to know what it means to say a matrix is invertible.)