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MATH 141: Linear Analysis I Homework 09 solved

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1. (if you did not finish this last week) T : R
n → R
m is a linear transformation.
(a) Is ker(T) a subspace of R
n?. Explain your reasoning. If yes, how can you find a basis for ker(T)?
(b) Is range(T) a subspace of R
m?. Explain your reasoning. If yes, how can you find a basis for range(T)?
(Hint: Connect ker(T) and range(T) to column space and nullspace of some matrix.)
2. A7×5 is matrix with 7 rows and 5 columns. The columns of A satisfy
(column-3) = −5(column-2) + (column-4).
Write down one concrete vector in N(A). Explain your reasoning.
3. In class, we agreed that N(A), the set of all solutions to A~x = ~0, is a vector subspace. What about all
solutions to an inhomogeneous system? More precisely, given a fixed matrix Am×n and fixed right hand
side vector ~0 6= ~b in R
m, define
V = {all solutions to A~x = ~b}.
Is V a vector subspace of R
n?
4. In class, we discussed to think of matrix multiplication as composition of functions. We thought about
how, if you have (AB)~x, where A and B are matrices and ~x is a vector, (AB)~x = A(B~x) could be thought
of as B transforming ~x first, and then A transforming the result of B~x.
© IOLA Team – iola.math.vt.edu
3. In class, we discussed matrix multiplication, when wanting to consider what happens to vectors or
the whole space under the transformations, as composition of functions. We thought about how, if
you have ABx, where A and B are matrices and x is a vector, ABx could be thought of as B
transforming x first, and then A transforming the result of Bx.
Now consider the composition of real-valued functions, familiar from high school. Use the given
functions f, g, and h below and carry out the requested computations.
a. ! ! ! and !(! ! )
b. ! ! 2 and !(! 2 )
c. ℎ(! ! )
d. ℎ(! 0 ) and ℎ(! −1 )
e. !(! ℎ ! )
f. Reflect on how composing these functions and evaluating at a value (such as at 2 in part b)
is similar to what we saw with matrix multiplication through the Pat & Jamie Task on
Wednesday. Write at least two thoughtful sentences.
4. Let the matrices F and G be defined as below. Answer the following questions accordingly.
! =
1 0 2
2 −1 0
0 3 4
, ! =
2 4 1
0 3 −2
5 0 1
a. Let ! =
1
2
−1
, and let Gx = y. Compute Gx and compute Fy.
b. Let
1
2
−1
, and let Fx = u. Compute Fx and compute Gu.
c. Explain how parts a) and b) illustrate matrix multiplication as composition of functions.
d. Explain how parts a) and b) illustrate that matrices F and G are not commutative.
5. Let A be a 5 x 4 matrix and B be a 4 x 3 matrix.
a. What is the domain of the transformation defined by B? What is the codomain of the
transformation defined by B?
b. What is the domain of the transformation defined by A? What is the codomain of the
transformation defined by A?
c. What is the domain of the transformation defined by AB? What is the codomain of the
transformation defined by AB?
d. Explain why the product BA is not defined. Ground your explanation in the concepts of
composition of functions, domain, and codomain.

f (x ) = 2x + 4 g (x ) = x
2 − 3x h(x ) = x
x − 2


x

B

x

A(B

x )
B A

A(B

x )
Transformed by B first
The result transformed by A
This exercise reinforces that connection.
(a) Given functions f and g below
f(x) = 2x + 4 g(x) = x
2 − 3x
compute
i. f(g(x)) and g(f(x))
ii. f(g(2)) and g(f(2))
(b) Let the matrices F and G be defined as below. Answer the following questions accordingly.
F =


1 0 2
2 −1 0
0 3 4

 G =


2 4 1
0 3 −2
5 0 1


i. Let ~x =


1
2
−1

, and let G~x = ~y. Compute G~x and compute
MATH 141: Linear Analysis I Homework 09
ii. Let ~x be the same vector as in i., and let F ~x = ~u. Compute F ~x and compute G~u.
iii. Compute F G and GF.
(c) Summarize, in words, the similarities between matrix multiplication and composition of functions.
Point out the equivalence, in terms of compositions of functions f and g, of the various quantities in
part (b): G~x, F ~y, F ~x, G~u, F G and GF. All notations have the same meaning as in parts (a) and (b).
(d) Is matrix multiplication commutative (That is, AB = BA for any matrices A and B)? Why or why
not?
When we solved the Italicizing N Task 1 problem in class, some groups have written all the input vectors side
by side into a matrix and all the output vectors the same way:

1 1/3
0 4/3
 0 −2 0
3 3 3
=

1 −1 1
4 4 4
We used that as an example to introduce one interpretation of matrix multiplication—-each column of the product matrix AB is the product of matrix A with the corresponding column vector of matrix B. Keep this interpretation in mind when answering following questions.
5. In order for us to be able to multiply two matrices A and B together, what conditions do we have to put
on the shapes of A and B? What is the shape of the product matrix AB?
6. (a) Fill in the blanks and explain your reasoning: Each column vector of the product matrix AB is
a linear combination of , and so each column vector of AB is in the span of
.
(b) As a consequence of part (a), what can you say about the relation among three column spaces C(AB),
C(A), and C(B)?
7. Assume that AB is defined. Determine the following statements true or false. If true, provide a justification. If false, provide a counterexample.
Hint: You may start by applying the definition of (in)dependence to columns of A or B and then try to
multiply the equation by the other matrix.
(a) If the columns of B are linearly dependent, then so are the columns of AB.
(b) If the columns of A are linearly independent, then so are the columns of AB.
8. True of false? If true, explain why. If false, provide a counterexample. A and B are matrices of appropriate
shape so that each addition or multiplication is defined.
(a) If columns 1 and 3 of B are the same, so are columns 1 and 3 of AB.
(b) If AB and BA are defined then A and B are square.
(c) If AB and BA are defined then AB and BA are square.
(d) (AB)
2 = A2B2
.
(e) (A + B)
2 = A2 + 2AB + B2
(f) If AB = B then A = I.
9. (Strang, §1.6, #25) Suppose that A is a 3×3 matrix with (Row-1)+(Row-2)=(Row-3).
(a) Explain why A~x =


1
0
0

 cannot have a solution.
MATH 141: Linear Analysis I Homework 09
(b) Which right-hand sides ~b =


b1
b2
b3

 might allow a solution to A~x = ~b?
(c) What happens to Row-3 if we perform forward elimination on A?
(d) Explain why each of the above three situations leads to the conclusion that A is not invertible? (Hint:
Think in terms of the linear transformation LA that A defines.)
10. (Strang, §1.6, #40) True or False. If true, explain why. If false, show a concrete counterexample. (Hint: Use
the fact that A is invertible if and only if the linear transformation LA which it defines is invertible.)
(a) A 4×4 matrix with a row of zeros is not invertible.
(b) A matrix with 1’s down the main diagonal is invertible.
(c) If A is invertible, then A−1
is invertible.