Ann has 8 apples she give 2 to Jack how much apes does he have left?

# Mathematics

Instructions

• The “Calculations” column needs to show each step that you would need to write out if you were showing your work when doing this problem by hand or trying to teach the concept to a friend.

• For each step, you will also have to provide a thorough a description of your thought processes for the calculation in the “Explanations” column.

• The first row (or step) should always be to copy the entire problem into the “Calculations” column or write out the important information. You can use copy and paste.

• Make sure that your final answer is clearly stated.

• You do not have to use all of the rows provided. However, if you need more rows to write out all of the steps you took to complete the problem, add rows.

W3.1

1 1 in = 2.54cm Copy the problem or write out important information.

2

(Add rows as needed).

Find the circumference of the circle.

B.) express the answer, rounding to the nearest tenth. Use the button on your scientific calculator.

1 Copy the problem or write out important information.

2

(Add rows as needed).

B.) express the answer, rounding to the nearest tenth. Use the button on your scientific calculator.

Click Here To Get More On This Essay!!!

1 Copy the problem or write out important information.

2

(Add rows as needed).

# MATHEMATICS

MATHEMATICS

ALGEBRA

1

:

(a) Prove that that there is no linear transformation

F

:

R

3

!

R

3

such that ker(

F

) =

R

(

F

).

(b) Find a linear transformation

F

:

R

2

!

R

2

such that ker(

F

) =

R

(

F

). You must explicitly

show that your example has this property.

2

:

Let

F

:

R

3

!

R

2

be the linear transformation

F

(

x;y;z

) = (

x;y

+

z

) and let

G

:

R

2

!

R

3

be

dened by

G

(1

;

0) = (0

;

1

;

2) and

G

(0

;

1) = (0

;

2

;

0).

(a) Find the standard matrix of the composition

G

F

.

(b) Find the kernel and range of

G

F

.

(c) Find the standard matrix of the composition

F

G

.

(d) Find the kernel and range of

F

G

.

3

:

Let

F

:

R

2

!

R

2

be (anticlockwise) rotation by

3

4

, let

G

:

R

2

!

R

2

be the orthogonal projection

onto the line

y

=

3

x

, and let

H

:

R

2

!

R

2

be a dilation by a factor of

p

8 (i.e. a dilation with

t

=

p

8). Find the standard matrix for each of the following linear transformations:

(a)

F

H

G

(b)

F

G

H

(c)

H

G

F

.

4

:

(a) Find the standard matrix

A

of the linear transformation

F

representing the cyclic permu-

tation

F

(

x

1

;x

2

;x

3

;x

4

) = (

x

4

;x

1

;x

2

;x

3

)

:

(b) What is the eect of the composition

F

F

? What is the standard matrix for

F

F

?

(c) Show that

A

3

=

A

1

. Why would you have expected this for the standard matrix of

F

?

(d) Is

F

invertible? Explain.

5

:

Let

F

:

R

l

!

R

m

and

G

:

R

m

!

R

n

be linear transformations.

(a) Show that ker(

F

)

ker(

G

F

).

(b) Show that

R

(

G

F

)

R

(

G

).

6

:

Classify the following conic sections by completing the square to put them into standard form

and sketch each one:

(a) 4

x

2

+

y

2

20

x

+ 24 = 0

(b)

x

2

+

y

2

2

x

+ 2

y

+ 2 = 0

(c)

y

2

+ 3

x

+ 6

y

+ 8 = 0

(d)

4

x

2

+ 9

y

2

4

x

6

y

+ 3 = 0

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