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

    1. 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.)

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

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

      1. in terms of pivots of A;

      1. in terms of the column vectors of A;

  1. 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:

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

    1. 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.

  1. 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(LA).

  1. of matrix A. Describe range(LA).

MATH 141: Linear Analysis I Homework 06

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

  1. 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.

  1. 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 LA.

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

(b) ker(LA) is another name for

(c) range(LA) is another name for

      • 3

2

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

  1. Let Am n by an m-by-n matrix and LA : Rn ! Rm 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. For the A and LA from the previous problem, is LA one-to-one? Is LA onto?

  1. (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 LA : Rn ! Rm by LA(~x) = A~x.

    1. Write down equivalent statements to

LA is one-to-one”

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

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

    1. in terms of the column vectors of A;

    1. in terms of pivots in A.

  1. Write down equivalent statements to

LA is onto”

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

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

  1. in terms of the column vectors of A;

  1. in terms of pivots in A.

.


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