# 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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