Let T: P2 → P3 be the transformation that maps a polynomial p(t) into the polynomial (t-2)p(t).
a. Find the image of p(t) = 2-t + t^2
b. Show that T is a linear transformation.
c. Find the matrix for T relative to the bases (1, t,t^2) and (1,t,t^2,t^3).
(a) Applying T to p(t) = 2 - t + t ² gives
T ( p(t) ) = (t - 2) (2 - t + t ²) = -4 + 4t - 3t ² + t ³
(b) T is a linear transformation if for any p(t) and q(t) in P₂ and complex scalars a and b, the image of any linear combination of p and q is equal to the linear combination of the images of p and q. In other words,
T ( a p(t) + b q(t) ) = a T ( p(t) ) + b T ( q(t) )
Let
p(t) = α₀ + α₁ t + α₂ t ²
q(t) = β₀ + β₁ t + β₂ t ²
Compute the images of p and q :
T ( p(t) ) = (t - 2) (α₀ + α₁ t + α₂ t ²)
… = -2α₀ + (α₀ - 2α₁) t + (α₁ - 2α₂) t ² + α₂ t ³
Similarly,
T ( q(t) ) = -2β₀ + (β₀ - 2β₁) t + (β₁ - 2β₂) t ² + β₂ t ³
Then
a T ( p(t) ) + b T ( q(t) ) = a (-2α₀ + (α₀ - 2α₁) t + (α₁ - 2α₂) t ² + α₂ t ³) + b (-2β₀ + (β₀ - 2β₁) t + (β₁ - 2β₂) t ² + β₂ t ³)
… = c₀ + c₁ t + c₂ t ² + c₃ t ³
where
c₀ = -2 (a α₀ + b β₀)
c₁ = a (α₀ - 2α₁) + b (β₀ - 2β₁)
c₂ = a (α₁ - 2α₂) + b (β₁ - 2β₂)
c₃ = a α₂ + b β₂
Computing the image of a p(t) + b q(t) would give the same result; just multiply it by t - 2 and expand. This establishes that T is indeed linear.
(c) Find the image of each vector in the basis for P₂ :
T (1) = (t - 2) × 1 = t - 2
T (t ) = (t - 2) t = t ² - 2t
T (t ²) = (t - 2) t ² = t ³ - 2t ²
Then
[tex]T=\begin{bmatrix}-2&0&0\\1&-2&0\\0&1&-2\\0&0&1\end{bmatrix}[/tex]
A rectangular based prism has a length of
12m a width of 7m and height of 11m. Find
its volume.
Answer:
924m^3 is the volume of the rectangular prism
how many solutions does 4x8y=-8 x=-2y+1 have
Answer:
It has no solution
Step-by-step explanation:
I tried to figure it out but I got no solution
Pythagorean Theorem Quiz.
1.) Find the unknown side length. Simplify answers that are radicals.
4
3
2.) Find the unknown side length. Simplify answers that are radicals.
16
12
Answer:
1.) 5 2.) 10.58
Step-by-step explanation:
For each one, you use the equation a^2 + b^2 = c^2 where c is the "hypotenuse"-- the side of the triangle opposite (facing) the right angle which is shown with a small box. The variables a and b do not matter, as long as they are the "legs", the other 2 sides.
So it is 4^2 + 3^2 = c^2 which simplifies to 16+9=25=c^2 and square root both sides and you get c=5
and 12^2 + b^2 = 16^2 which simplifies to 144+ b^2 = 256 and subtract 144 from both sides and you get b^2 = 112 and square root both sides and it is approximately 10.6 :)
The ratio of the measures
of the angles of a triangle is
6:10:2. What is the measure of
the smallest angle.
Answer:36
Step-by-step explanation:
You would add 6x + 10x + 2x = 180 then add your xs to get 18x =180 divide to 18 then multiply by 2 and get 35 :)
What is the measure of an interior angle of a regular 10-gon?
Answer:
Step-by-step explanation:
Work out the area of the trapezium ABDE.
Answer:
(half * base 1 * base 2)/height
base 1 is 6 and base 2 is 9
get the height from similarity
6/9 = 8/CE
therefore CE = 72/6
Answer:
30cm²
Step-by-step explanation:
AE/BD = CE/CD
9/6 = CE/8
CE = 9/6 x 8
CE = 12cm
DE = CE - CD
DE = 12 - 8 = 4 cm
½(6 + 9)4 = 30cm²
what is 3/5 divide 1/3
Answer:
1 4/5
Step-by-step explanation:
Henry went clothes shopping and bought 3 pairs of shorts for $22.99 each, 2
shirts for $32.99 each and one pair of shoes for $54.99. What is the most
reasonable estimate of the total amount he spent on all the clothing?
A. $135
B. $184
C. $190
D. $223
Answer:
C
Step-by-step explanation:
Please help me answer this :)
Explain your answer
Will give brainiest ! Oh
Answer:
Step-by-step explanation:
Answer:
C
Step-by-step explanation:
Because the points are going up by 10 :)
PLEASE ANSWER GIVING 30 POINTS
Answer:
this is only 15points but ok
Step-by-step explanation:
answer: D)
Answer:
The answer should be 15.
Step-by-step explanation:
[tex]\sqrt[3]{125}[/tex] Factor 125 = [tex]5^{2}[/tex] x 5. Rewrite the square root of the product [tex]\sqrt{5^{2}X5 }[/tex] as the product of square roots. [tex]\sqrt{5^{2} }[/tex] [tex]\sqrt{5}[/tex]. Take the square root of [tex]5^{2}[/tex].
[tex]3[/tex] x [tex]\sqrt[5]{5}[/tex] Multiply 3 and 5 to get 15.
[tex]\sqrt[15]{5}[/tex]
-5/12 - (-9/3)=
Help me please
Answer:
HOW T.F DO YOU NOT
Step-by-step explanation:
-5/12+3
2 7/12
UST CAUSE
Answer: -5/4
Step-by-step explanation:
6) A right triangle its rotated about an axis that passes through its height as shown. Which type of object will be formed by its rotation? Describe in words or draw a picture
object will be formed by its rotation? Describe in words or draw a picture.
Answer:
a cone
Step-by-step explanation:
i got it right
Answer:
CONE
Step-by-step explanation:
Which expression would be easier to simplify if you used the associative property to change the grouping?
A.(160+40)+27(160+40)+27
B.0.85+[0.15+(-\frac{1}{3})]0.85+[0.15+(− 31 )]
C.[(-\frac{1}{3})+(-\frac{2}{3})]+(-\frac{4}{5})[
D.1 + ( 3/4 + ( - 3/4)
brainliest if right and its on ap..ex

Answer:
C is the answer
Step-by-step explanation:
Somehow this was in my test last week , lol
If 4 movie tickets cost $28, how much would it cost to buy 9 movie tickets
Answer:
$63
Step-by-step explanation:
First, divide 28 by 4 to get 7
Now we know that one ticket cost $7.
Do 7 times 9 to get 63
9 tickets costs $63
9 movie tickets would cost $63.
Since 4 movie tickets cost $28, you can divide 28 by 4 to get 7. From here, you can multiply 7 by 9 to get 63.
May I have brainliest please? :)
Can someone please help me Please I'm desperate
Rob is 7 years older than James. Kate is 2 years younger than James. If the sum of their ages is 116, how old are each of the siblings?
Answer:
James is 37, Rob is 44, and Kate is 35
Step-by-step explanation:
Set up an equation where x is James' age:
Rob's age can be represented by x + 7, since he is 7 years older than James
Kate's age can be represented by x - 2, since she is 2 years younger than James
Add these together in the equation and set it equal to 116:
(x + 7) + (x - 2) + (x) = 116
Add like terms and solve for x:
(x + 7) + (x - 2) + (x) = 116
3x + 5 = 116
3x = 111
x = 37
So, James is 37.
Find Rob's age by adding 7 to this:
37 + 7
= 44
Find Kate's age by subtracting 2:
37 - 2
= 35
So, James is 37, Rob is 44, and Kate is 35
Answer:
Rob = 44
Kate = 35
James = 37
Step-by-step explanation:
Rob = 44 which is 7 years older than James
Kate = 35 which is 2 years younger than James
James = 37
To catch a ball, a professional baseball player leaps into the air with an initial velocity of
14 feet per second. Write a model for the height of the player above the ground after t seconds. After how many seconds does the player land on the ground? What would be the domain of this function?
Answer:
The answer is below
Step-by-step explanation:
a) The baseball player leaps into the air with an initial velocity of 14 feet per second. This means that the player leaps from the ground, the initial height is therefore zero.
The height (y) is given by the formula:
y(t) = ut - (1/2)gt² + initial heigth
u = initial velocity = 14 ft/s, t = time taken, g = acceleration due to gravity = 32 ft/s². Substituting:
y(t) = 14t - (1/2) * 32 *t² + 0
y(t) = 14t - 16t²
b) when the player is on the ground, the height = 0. hence:
0 = 14t - 16t²
16t² - 14t = 0
t(16t - 14) = 0
t = 0 or 16t - 14 = 0
t = 0 or 16t = 14
t =0 or t = 0.875
Hence t = 0.875 seconds
c) The domain is the set of possible values for the time. Hence:
Domain = (0, 0.875] = 0 < t ≤ 0.875
9. A cell phone can be purchased from the manufacturer for $160. The online retailer has a
standard markup of 30%, and the store has a standard markup of 40%.
(a)What is the price of the cell phone when purchased online?
(b)What is the price of the cell phone when purchased at the store?
(c) Which is the better deal?
Answer Below
30% of 160 is 48 dollars. Meaning you would take the 48 from 160. Leaving 112.So the 30% mark up would mean you would pay $112.Saving $48.
Take the 160 dollar,and have the 40% to work with. Taking this would be $64. So 160-64 is $96. They will spend $96 on the phone in-store
Purchase online- $112
Purchase in-store-$96
The better deal is in-store
Answer:
(A) $208 (B) 224 (C) the online one. it cost less
Step-by-step explanation:
(A) find 30% of $160 and add it to 160. put that answer in dollars.
(B) find 40% of $160 and add it to 160. put that answer in dollars.
(C) it is the one that costs less
Please Help ASAP!! ❤️
Answer:
Step-by-step explanation:
*a is 6:55
*b is 37
*c is 15:12
The sum of three angle measures in a triangle is always 180°.
Answer:
YES IT IS RIGHT.......!
Look at pic and pls halp
Identify the interval(s) on which the function y = 2x^2 - 8x - 10 is positive.
A. -1
B. x < -1 and x > 5
C. -5
D. X<-5 and x > 1
Given:
[tex]y=2x^2-8x-10[/tex]
To find:
The interval(s) on which the given function is positive.
Solution:
We have,
[tex]y=2x^2-8x-10[/tex]
[tex]y=2(x^2-4x-5)[/tex]
[tex]y=2(x^2-5x+x-5)[/tex]
[tex]y=2(x(x-5)+1(x-5))[/tex]
[tex]y=2(x+1)(x-5)[/tex]
For zeroes, [tex]y=0[/tex]
[tex]2(x+1)(x-5)=0[/tex]
[tex]x=-1,x=5[/tex]
-1 and 5 divide the number line in three parts [tex](-\infty , -1),(-1,5),(5,\infty )[/tex].
Interval Selected Value [tex]y=2(x+1)(x-5)[/tex] Sign
[tex](-\infty,-1)[/tex] -2 [tex]y=2(-2+1)(-2-5)=14[/tex] +
[tex](-1,5)[/tex] 0 [tex]y=2(0+1)(0-5)=-10[/tex] -
[tex](5,\infty)[/tex] 6 [tex]y=2(6+1)(6-5)=14[/tex] +
So, the function is positive for interval [tex](-\infty,-1)[/tex] and [tex](5,\infty)[/tex].
We can say that the function is positive for interval x < -1 and x > 5.
Therefore, the correct option is B.
Provide a radian measure of an angle that is 5/8 of a revolution counterclockwise from the x-axis.
You are your friend are playing a game where you flip a coin and roll a fair four-sided die.
What about it?????????
Answer: is this the best ratio question
Step-by-step explanation:
Totally blanking on how medians work
===============================================
The three medians of this triangle are:
PFREQDThese segments go from one vertex to the midpoint of the opposite side.
The median we'll focus on is PF.
The point C is the centroid of the triangle. It's where the three medians intersect. It turns out that C divides PF such that CF is twice as long as PC
In other words,
CF = 2*PC
This means,
PF = PC+CF .... segment addition postulate
PF = PC+2*PC ... replace CF with 2*PC
PF = 3*PC .... combine like terms
So the median PF is three times the length of its portion PC
We're told that PF = 24
We can then find the following:
PF = 3*PC
24 = 3*PC
3*PC = 24
PC = 24/3
PC = 8
Then we double this to get the length of CF
CF = 2*PC
CF = 2*8
CF = 16
This is the same as FC because the order of the endpoints don't matter when it comes to naming a segment.
The final answer is 16.
If there were 1,593 Ronald McDonald's in the McDonald's, how many could fit?
Answer:
over 1,593
Step-by-step explanation:
if there was already 1,593 Ronalds in the McDonalds the McDonalds can most likely fit over that amount
Markers were placed at altitudes of 10 and 15 feet below sea level Enter the integer that represents the altitude closer to sea level
Answer:
43
neeed...20...charetors
The integer that represents the altitude closer to sea level is 10.
What is addition?Addition is one of the mathematical operations. The addition of two numbers results in the total amount of the combined value. A numerical expression is algebraic information stated in the form of numbers and variables that are unknown.
We are informed that Markers were placed at altitudes of 10 and 15 feet below sea level
Then we need to find the integer that represents the altitude closer to sea level.
We can say that 10 is negative (below the sea level), and 10 is 10 spaces away from zero on the number line, meaning the answer could be |10|.
The sea level would be 0 and the 10 would be -10 and the 15 will be -15
Hence, the 10 is closer to the sea level.
Thus the integer that represents the altitude closer to sea level is 10,
Learn more about fraction addition here:
https://brainly.com/question/17544795
#SPJ2
16 – (3 – 4y) = 2(y + 2)
Answer:
[tex]y = -4\frac{1}{2}[/tex]
Step-by-step explanation:
[tex]16-(3-4y) = 2(y+2)[/tex]
Distribute.
[tex]16 - 3 + 4y = 2y + 4[/tex]
Combine like terms.
[tex]13 + 4y = 2y + 4[/tex]
Subtract [tex]2y[/tex] from both sides.
[tex]13 + 2y = 4[/tex]
Subtract 13 from both sides.
[tex]2y = -9[/tex]
Divide both sides by 2.
[tex]y = -4\frac{1}{2}[/tex]
I hope this helps!
11.) Is the equation below linear or nonlinear?
Y=9x-5
Answer:
liner
Step-by-step explanation:
There are 23 students in a sixth-grade class. Nine students play hockey.
What is the ratio of students who do not play hockey to the total amount of
students? (6.RP.1)
point
O 9 to 23
O 14 to 23
O 14 to 9
O 9 to 14
Answer:
14:23
Step-by-step explanation:
14:23
Answer:
its 14-23
Step-by-step explanation: