MATH141-Homework 6 Solved

1.   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.

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

(d)    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.)

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

(a)    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:

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

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;

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:

(a)    once in terms of pivots of the matrix A, and

(b)    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.

(a)    The first and third columns of U are linearly independent.

(b)    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



(c)     The first and third columns of the original matrix A are linearly independent.

(d)    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.

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

6.   Let and denote the function it defines as LA. That is, LA : Rn → Rm, LA(~x) = A~x.

Answer the following questions about this particular LA.

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

(b)    ker(LA) is another name for       of matrix A. Find ker(LA).

(c)     range(LA) is another name for    of matrix A. Describe range(LA).

(d)    Find the image under . Find all vectors ~x’s who have the same LA(~u) as its image.

7.   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.

(c)     For the A and LA from the previous problem, is LA one-to-one? Is LA 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 LA : Rn →Rm by LA(~x) = A~x.

(a)    Write down equivalent statements to

”LA is one-to-one”

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

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

iii.    in terms of the column vectors of A;

iv.    in terms of pivots in A.

(b)    Write down equivalent statements to

”LA is onto”

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

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

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