[Solved] MATH141: Linear Analysis I Homework 06

1. (if you have not done this problem from last week) (Strang 2.2 #39) Explain why all these statements are all false (all statements are about solving linear systems A~x =^~b):
   • The complete solution is any linear combination of ~x_particular and ~x_nullspace.
   • A system A~x =^~b has at most one particular solution.
   • The solution ~x_particular with all free variables zero is the shortest solution (minimum length k~xk). (Find a 22 counterexample.)
   • If A is invertible there is no solution ~x_nullspace in the nullspace. (Lei Yues 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)
   • Write equivalent statements of the sentence:

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

Explain in each case why your statement is equivalent.

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

• 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^~bs.

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

A~x =^~b is solvable for any^~b.

Explain in each case why your statement is equivalent.

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

• in terms of the column vectors of A;

3. Complete the worksheet titled Existence and Uniqueness of Solutions. Study your examples, and summarize the method to come up with examples satisfying each pair of criteria twice:
   • once in terms of pivots of the matrix A, and
   • 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.
5. A is a 3-by-4 matrix and its upper echelon form is. Determine the following state-

ments true or false. Explain your reasoning.

• The first and third columns of U are linearly independent.
• The second column of U is a linear combination of its first and third columns. So is the fourth column of U.

MATH 141: Linear Analysis I Homework 06 Fall 2019

• The first and third columns of the original matrix A are linearly independent.
• 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.
• A and U have the same column space. That is, C(A) = C(U).

6. Let and denote the function it defines as L_A. That is, L_A : Rⁿ R^m, L_A(~x) = A~x.

Answer the following questions about this particular L_A.

• What are the values of m and n?
• ker(L_A) is another name for of matrix A. Find ker(L_A).
• range(L_A) is another name for of matrix A. Describe range(L_A).
• Find the image under. Find all vectors ~xs who have the same L_A(~u) as its image.

7. Let A_mn by an m-by-n matrix and L_A ^: Rⁿ R^m the function it defines. Complete the following sentences and explain your reasoning.
   • L_A is onto if and only if range(L_A) .
   • L_A is one-to-one if and only if ker(L_A) . Hint: You may find problem#4 of Homework05 helpful.
   • 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 mn matrix and define L_A : Rⁿ R^m by L_A(~x) = A~x.

• Write down equivalent statements to

L_A is one-to-one

1. in terms of the existence and/or uniqueness of solutions;
2. in term of nullspace or column space of A;

• in terms of the column vectors of A;

1. in terms of pivots in A.

• 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;

• in terms of the column vectors of A; iv. in terms of pivots in A.

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