Homework 06 Solution

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1. (if you have not done this problem from last 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 ~xparticular and ~xnullspace.
(b)	~
	A system A~x = b has at most one particular solution.

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

1. 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.)

2. (Making connections of different perspectives of the same idea)

2. 1. Write equivalent statements of the sentence:

A~x = ~0 has only the ~x = ~0 solution.

Explain in each case why your statement is equivalent.

i. in term of N(A) or C(A);

ii. in terms of pivots of A;

iii. in terms of the column vectors of A;

~ ~

iv. in terms of the existence and/or uniqueness of solutions to A~x = b for other b’s.

(b) Write equivalent statements of (in other words, necessary and sufficient conditions to) the sentence:

~ ~

1. 1. 1. in term of N(A) or C(A);

1. 1. 2. in terms of pivots of A;

1. 1. 3. in terms of the column vectors of A;

3. Complete the worksheet titled ”Existence and Uniqueness of Solutions”. Study your examples, and sum-marize the method to come up with examples satisfying each pair of criteria twice:

3. 1. once in terms of pivots of the matrix A, and

3. 2. another time in terms of values of m, n, and r, where m is the number of rows of A, n the number of columns, and r = rank(A). Recall that rank(A) is, by definition, the number of pivots of A.

4.	~
	Do you think the set of all special solutions to A~x = 0 are linearly dependent, independent. or cannot be
	decided (meaning that special solutions to certain homogeneous systems are dependent while to others
	are independent)? Explain your reasoning.
		2	3	1	0	1	3. Determine the following state-
5.	A is a 3-by-4 matrix and its upper echelon form is U =		0	0	7	2
	ments true or false. Explain your reasoning.	4	0	0	0	0	5

1. The first and third columns of U are linearly independent.

2. The second column of U is a linear combination of its first and third columns. So is the fourth column of U.

1. of matrix A. Find ker(L_A).

2. of matrix A. Describe range(L_A).

MATH 141: Linear Analysis I Homework 06

3. The first and third columns of the original matrix A are linearly independent.

3. The second column of the original matrix A is a linear combination of its first and third column. So is the fourth column of A.

3. A and U have the same column space. That is, C(A) = C(U).

6. Let A =	1	5	7	and denote the function it defines as LA. That is, LA : Rn ! Rm, LA(~x) = A~x.
	3	7	5

Answer the following questions about this particular L_A.

(a) What are the values of m and n?

(b) ker(L_A) is another name for

(c) range(L_A) is another name for

1. 3. • 3

2

1. 3. Find the image under L_A of ~u = 4 1 5. Find all vectors ~x’s who have the same L_A(~u) as its image. 1

7. Let A_{m n} by an m-by-n matrix and L_A : Rⁿ ! R^m the function it defines. Complete the following sentences and explain your reasoning.

(a) LA is onto if and only if range(LA)			.
(b) LA is one-to-one if and only if ker(LA)				. Hint: You may find problem#4 of Homework05
helpful.

1. 3. For the A and L_A from the previous problem, is L_A one-to-one? Is L_A onto?

8. (making connections) Use the previous two problems as hint to write down the more general statements in this problem.

Let A be an m n matrix and define L_A : Rⁿ ! R^m by L_A(~x) = A~x.

8. 1. Write down equivalent statements to

”L_A is one-to-one”

1. 1. in terms of the existence and/or uniqueness of solutions;

1. 2. in term of nullspace or column space of A;

1. 3. in terms of the column vectors of A;

1. 4. in terms of pivots in A.

2. Write down equivalent statements to

”L_A is onto”

1. in terms of the existence and/or uniqueness of solutions;

2. in term of nullspace or column space of A;

3. in terms of the column vectors of A;

4. in terms of pivots in A.

.