(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]
Find the value of x
Answer:
x=20°
Step-by-step explanation:
The triangle is equilateral. (All sides are the same).
Each angle in an equilateral triangle is 60.
3x=60
x=20°
please help me with this
Is the following table proportional or non-proportional?
Answer:
proportional
Step-by-step explanation:
y= -7x
Marcus can drive his boat 36 miles down the river in 3 hours but takes 4 hours to return upstream. Find the rate of the boat
in still water and the rate of the current.
Answer:
With river 12 miles per hour
Against river 9 miles per hour
wv = water velocity vsw = boat velocity in still water
vsw +wv = 12 mph
vsw -wv = 9 mph
2 * vsw = 21 mph
boat velocity in still water = 10.5
Since vsw + wv = 12 then
water velocity = 1.5 mph
Step-by-step explanation:
Answer:
speed in still water: 10.5 Current: 1.5
Step-by-step explanation:
Find the value of x. 1) m angle2=3x-6 120° 2 B) 10 D) 6 A) 12 C) -11
Write the slope intercept form of the equation of the line through the given points
Answer:
9) y = -3x - 4
10) y = -1/3x + 14/3
11) y = x - 1
12) y = -7/5x - 18/5
13) y = 3/5x + 9/5
14) y = -2x - 3
Step-by-step explanation:
For this explanation, let's use the last problem as the example. You would use the formula y=mx+b. The first thing you would need to find would be the slope, or m. So, you would find the slope and conclude the answer is -2. After that, you would solve for b and get the answer. Hope this helped!
Find (12y + 20y2 – 2) + (-13y2 + y - 10). Write your answer in
standard form.
Answer:
7y²+13y-12
Step-by-step explanation:
20y²+12y-2-13y²+y-10
7y²+13y-12
Find the slope of the line.
please help
will mark brainlyest
Answer:
m = 1
Step-by-step explanation:
slope = (y₂-y₁)/(x₂-x₁)
m = (4 - -1) / 1 - -4) = 5/5 = 1
WILL GIVE BRAINLIEST
Find the equation of the line
perpendicular to y=-2x + 1 that
also intersects the point (8, 2).
y- Hx+ 1
Enter
Answer:
y=1/2x-2
Step-by-step explanation:
perpendicular lines have a neg. reciprocal for the slope.
for example:
in this equation y=-2x+1
-2 is the slope and the negative reciprocal would be 1/2
hence, the slop of our new line is 1/2
y=1/2x+b
we can substitute the coordinates given for x and y:
2=1/2(8)+b
2=4+b
-2=b
hence the new equation is y=1/2x-2
What is the answer to this question?
Answer:
the answer is the top right option
Step-by-step explanation:
SOLVE FOR X. pls i need help
I have 3 green cubes, 4 red cubes, 2 black cubes, and 6 yellow cubes. What is the probability that I will randomly choose a black cube, put it back then randomly choose a yellow cube
Answer:
4/75 or 0.05333
Step-by-step explanation:
3+4+2+6=15 elements
2/15= probability to choose a black
6/15 = probability to choose a yellow cube
Probability to choose a black AND (*) a yellow = 2/15*6/15=12/225=4/75 or 0.05333
A farm is to be built in the shape of Quadrilateral ABCD, as shown below. All four sides are equal.
A rhombus ABCD is shown with diagonal AC equal to 18. 3 feet and diagonal BD equal to 13.8 feet
What is the area of the farm? (1 point)
a
73.2 square feet
b
111.2 square feet
c
126.27 square feet
d
168.36 square feet
Answer:
c 126.27 square feet
Step-by-step explanation:
sum of two triangles
18.3 * (13.8/2) * 2 = 126.27
HELP FAST!!!! Please
A plane passes through this three-dimensional figure parallel to its base. Which shape represents the cross section that is created.
Answer:
c
Step-by-step explanation:
hope it helpp
Plz help! I really need this!
Answer:
x=16
Step-by-step explanation:
the remaining angle for thw triangle is found by subtracting the already shown angles from 180 (the angle some of a triangle) which happens to be 96 to get 84 but now you need to subtract 84 from 180 to find the angle on the other side of line because lines have a total angle measurement of 180 but we already kbow what this get's us because it's 96 now we devide 96 by 6 to get 16 our answer.
A rectangular mat has an area of 7/12 in2. The length of the mat is 5/6 in. What is the width of the mat
The width of the mat with the given area and length is 7/10 in.
What is the width of the mat?The area of a rectangle is expressed as;
A = l × w
Where l is length and w is width.
Given the data in the question;
Area A = 7/12 in²Length l = 5/6 inWidth w = ?We plug our given values into the above equation and solve for w.
A = l × w
7/12 = 5/6 × w
w = 7/12 ÷ 5/6
w = 7/12 ÷ 6/5
w = 42/60
w = 7/10 in
Therefore, the width of the mat with the given area and length is 7/10 in.
Learn more about area of rectangle here: https://brainly.com/question/20693059
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3(2ax-10)=24x+5b
work out the values of a and b
Answer:
a=4
b=-6
Step-by-step explanation:
3(2ax-10)=24x+5b
6ax - 30 = 24x + 5b
6a = 24
a = 4
5b = -30
b = -6
check: 3(2*4*x - 10) = 24x - 30 = 24x + 5 * (-6) = 24x - 30
I don’t think this is right I need help?
Answer:
1 and 2
Step-by-step explanation:
If x is 1: 2 x 1 + 7 = 2 + 7
= 9 This works.
If x is 2: 2 x 2 + 7 = 4 + 7
= 11 This works
If x is 3: 2 x 3 + 7 = 6 + 7
= 13 This does not work since 2x + 7 has to be less than 13.
the area of the figure
3x-2y=1x+4y=33 elimination process
Answer:
x = 5 ; y = 7
Step-by-step explanation:
3x - 2y = 1 --------------------(I)
x + 4y = 33 ------------(II)
Multiply equation (II) by (-3) and then add. So, x will be eliminated and we can find the value of y
(I) 3x - 2y = 1
(II)*(-3) -3x -12y = -99 {Now add}
- 14y = -98
y = -98/14
y = 7
Plugin y = 7 in equation (I)
3x - 2*7 = 1
3x - 14 = 1
3x = 1 + 14
3x = 15
x = 15/3
x = 5
Please help me asap this is due in 10 min Ill also mark brainliest
As we know that formula of area of circle is πr²
So let's apply formula
π×8.6×8.6
=232.2
=200 is the nearest hundredth.
You have to do rest of all through this method.
Beth put $8,000 in a savings account. At the end of a year
the account had earned $720 in interest. What was the
yearly interest rate on the account?
Answer:
It's %9.0.
Step-by-step explanation:
What is the probability of picking a yellow marble and a blue marble of the yellow marble is replaced ?
Answer:
50%
Step-by-step explanation:
How far is `10\%` of a `2,000`-kilometer trip?
determine the slope of the linear graph y=12x+7
Answer:
12x
Step-by-step explanation:
The graph y = 1/2x^2 - x - 4 is shown below. State the coordinate of the zero(s):
Answer:
Zeros → (-2, 0) and (4, 0)
Step-by-step explanation:
Equation of the graph has been given as,
y = [tex]\frac{1}{2}x^{2} -x-4[/tex]
For zeros of the function,
y = 0
Therefore, [tex]\frac{1}{2}x^{2} -x-4=0[/tex]
By solving the equation algebraically,
x² - 2x - 8 = 0
x² - 4x + 2x - 8 = 0
x(x - 4) + 2(x - 4) = 0
(x + 2)(x - 4) = 0
x = -2, 4
Therefore, coordinates of the zeros will be (-2, 0) and (4, 0).
Let's confirm the zeros (x-intercepts) from the graph attached,
Coordinates of the x-intercepts are,
(-2, 0) and (4, 0)
What is the constant of proportionality for the table? What does it represent? Reminder: k= y/x
Answer:
The constant of proportionality is 5. It represents the cups of yellow paint needed for 1 cup of blue paint.
what is the correct answer ro this problem? 121 ÷ 11 = ?
Answer:
11
Step-by-step explanation:
quik mafs
Guys help me please ,how to solve it? i need a solution
Answer:
well, x = 5
Step-by-step explanation:
here,
EB Is parallel to DC , so
Angle AEB = Angle ADE
( because they are corresponding angle pair )
Angle ABE = Angle ACD
( because they are corresponding angle pair )
triangle ADC Is similar to triangle AEB
(By AA similarity criteria )
hence AE / ED = AB / BC ( by CPCT )
so 24/10 = 12/x
=》 24/12= 10/x
=》 x = 5
okay i hope u got it
30 POINTS WOWWWWWW plz answer :)
Answer:
Step-by-step explanation:
