Homework 05 Solution

Description

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.

1. (Strang x1.2 #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)-

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 ^~, what is the coefficient matrix , what is

, and what is ^~?

~x b

2. What is the definition of the nullspace of a given matrix? Describe the nullspace N(A) of this partic-ular A as a span.

2. 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)			~	~	~				first
	Choose three concrete right-hand vectors b1			; b2; b3 for which the system is inconsistent. On your
	~	~	~
	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
			~	~	respectively. How many solutions are there for each
	solutions to the two systems for with b4 and b5
					~		~	~	and
	system? On your second graph, sketch N(A), C(A), b4						; b5	, and all solutions corresponding to b4

5. • _.

b₅

5. 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

4. (cf. Strang x1.3 #18) Suppose that ~x₀ and ~x₁ are two different solutions to the same linear system of equa-

tions ~, where is the coefficient matrix, the unknown, and ~ may or may not be the zero vector.A~x=bA~xb

(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?

3. (my own) Explain how this problem is related to the second sentence you need to complete in part (f) of problem 3.

(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 con-ditions 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.

1. 5. Find N(A).

1. 5. 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 per-form Gaussian elimination on a matrix A to get its upper echelon form U, are N(A) and N(U) differ-ent? Why or why not?

6. 2. Repeat the discussion above for column space.

6. (a) (Strang x2.2 #35) What conditions on b₁; b₂; b₃; b₄ make each system solvable? Solve for ~x.

22		43 x1			= 2b23				22		4	6 3		2	x2	3	= 2b23
	1	2		x2		b1				1	2	3			x1			b1
	2	5				b3				2	5	7						b3
6	3	9	7		6	b		7	6	3	9	12	7	4		5	6	b		7
															x3
6			7		6		4	7	6				7				6		4	7
4			5		4			5	4				5				4			5

(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 x2.2 #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 ~x_particular and ~x_nullspace.

MATH 141: Linear Analysis I Homework 05

(b) A system A~x = ^~b has at most one particular solution.

3. The solution ~x_particular with all free variables zero is the shortest solution (minimum length k~xk). (Find a 2 2 counterexample.)

3. If A is invertible there is no solution ~x_nullspace in the nullspace. (Lei Yue’s comment: you do not even need to know what it means to say a matrix is invertible.)