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(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 ~x_{particular} and ~x_{nullspace}.
(b)
~
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 2 2 counterexample.)



If A is invertible there is no solution ~x_{nullspace} in the nullspace. (Lei Yue’s comment: you do not even need to know what it means to say a matrix is invertible.)


(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.
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:
~ ~



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





in terms of pivots of A;





in terms of the column vectors of A;



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. 

2 
3 
1 
0 
1 
^{3}. Determine the following state 

5. 
A is a 3by4 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 

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.

of matrix A. Find ker(L_{A}).

of matrix A. Describe range(L_{A}).
MATH 141: Linear Analysis I Homework 06

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 A =
1
5
7
and denote the function it defines as L_{A}. That is, L_{A} : R^{n} ! R^{m}, L_{A}(~x) = A~x.
3
7
5
Answer the following questions about this particular L_{A}.
(a) What are the values of m and n?
(b) ker(L_{A}) is another name for
(c) range(L_{A}) is another name for



3


2


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


Let A_{m n} by an mbyn matrix and L_{A} : R^{n} ! R^{m} the function it defines. Complete the following sentences and explain your reasoning.

(a) L_{A} is onto if and only if range(L_{A})
.
(b) L_{A} is onetoone 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} onetoone? Is L_{A} onto?


(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 L_{A} : R^{n} ! R^{m} by L_{A}(~x) = A~x.


Write down equivalent statements to

”L_{A} is onetoone”


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



in term of nullspace or column space of A;



in terms of the column vectors of A;



in terms of pivots in A.


Write down equivalent statements to
”L_{A} is onto”

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

in term of nullspace or column space of A;

in terms of the column vectors of A;

in terms of pivots in A.
.