MATH 141: Linear Analysis I Homework 06 solved

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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):
(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 U =


3 1 0 −1
0 0 −7 2
0 0 0 0

. Determine the following statements 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 A =

1 −5 −7
−3 7 5 
and denote the function it defines as LA. That is, LA : R
n → R
m, 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 LA of ~u =


2
1
−1

. 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 : R
n → 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.
(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 : R
n → R
m 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.
.