(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]
The top of a 25-foot ladder is sliding down a wall at a rate of 3 feet per second. When the top of the ladder is 15 feet from the ground, what is the rate of change between the bottom of the ladder and the wall?
Answer:
1.8ft/s
Step-by-step explanation:
Using the formula for calculating Speed as shown:
Speed = Distance/Time
Time = Distance/Speed
t = d/s
Given
d1 = 25 foot
s1 = 3ft/s
d2 = 15ft
Required
s2
Using the formula;
d1/s1 = d2/s2
s2 = s1d2/d1
s2 = 3*15/25
s2 = 45/25
s2 = 1.8ft/s
Hence the rate of change between the bottom of the ladder and the wall is 1.8ft/s
When is subtraction always the first operation done in an expression? a. When it is the first operation in an expression, from left to right. b. When it follows addition in an expression. c. When it is in parentheses in an expression. d.Never. 1.
Answer:
Naruto usamaki is the greatest hokagey in the leaf village
Step-by-step explanation:
I took the test on edge
Can somebody help me on this ?
Answer:
drag the protactor so that the the bottom, middle line is centered in point B, and the bottom edge is flat against line BC
the angle will be whatever number AB lines up with on the outer ring (the big numbers)
Step-by-step explanation:
Help pls I have no idea lol
Answer:
a=7b=9c=4
Step-by-step explanation:
Out of 498 applicants for a job, 187 have over 10 years of experience and 127 have over 10 years of experience and have a graduate degree. Step 1 of 2 : What is the probability that a randomly chosen applicant has a graduate degree, given that they have over 10 years of experience
Answer:
the probability that randomly selected applicants over 10 years of experience is 0.6791
Step-by-step explanation:
The computation of the probability that randomly selected applicants over 10 years of experience is as follows:
Total would be
= 187 have 10 + years experience
Now
P(graduate | 10+) = (graduate and 10+ years experience) ÷ (10 + years of experience)
= 127 ÷ 187
= 0.6791
Hence, the probability that randomly selected applicants over 10 years of experience is 0.6791
please i need help:/
Find the area of triangle abc
Passes through (-4,-5) Slope = -314
Please Helppppp!!!!!!!!!!!!
Answer:
[tex]y=-314x-1261[/tex]
Step-by-step explanation:
When we are writing an equation we have to put it in slope-intercept-form [tex]y=mx+b[/tex]. We already know the slope it's -314, we replace m with the slope. Now we have to find b which is the y-intercept. So what we have to do is plug in the ordered pair into the equation. -4 for x and -5 for y.
[tex]y=mx+b\\-5=-314(-4)+b[/tex]
Now we solve
[tex]-5=-314(-4)+b\\-5= 1256+b\\b=-1261[/tex]
So now that we know everything we need we plug in and....
[tex]y=mx+b\\y=-314x-1261[/tex]
we've got our equation!
Answer:
The line slants upward, so the slope is positive.
What is 10% of 10$ plz
Answer:
1 dollar
Step-by-step explanation:
Answer:
1 dollar
Step-by-step explanation:
Hope this helps!
What involves examining and evaluating information, as well as weighing alternatives and considering consequences?
Answer: Critical Thinking.
Step-by-step explanation:
Critical thinking is the process of conceptualizing, applying, analyzing, synthesizing, and/or reviewing data acquired from, or generated by, observation, experience, reflection, reasoning, or communication actively and skillfully as a guide to intellectually disciplined belief and action.
Solve to find X. 3 ≥ -3x
Answer:
-1 ≤ x
Step-by-step explanation:
You just need to divide -3 on both sides.
-1 ≤ x
Note that since you are dividing by a negative number, the sign flips.
-1 ≤ x
Step-by-step explanation:
First start with the equation:
3 ≥ -3x
divide by -3 to get x positive and singular:
-1 ≥ x
and since we divided by a negative we need to flip the sign:
-1 ≤ x would be your answer
hope this helped :)
1. A 15-pound bag of sugar is being split into containers that hold of a pound. How many
containers of sugar will the 15-pound bag create?
The answer is: 17 1/7 containers. The work is on the photo
Answer:
yes
Step-by-step explanation:
yes
for each one fifth, use a horizontal line or lines to show fractions equivalent to 1/5 and write the equivalent fractions
Answer:
2/10
Step-by-step explanation:
PLEASE ANSWER THIS ASAP, I WILL BE GIVING YOU A THANKS! ..
Answer:
D
Step-by-step explanation:
7 - z = 3 + z + 4 - 2z
7 - 3 - 4 = z - 2z + z
0 = 0 (Infinitely many solutions)
5(a + 4) - 15 = -5(a - 1)
5a + 20 - 15 = -5a + 5
5a + 5a = 5 - 20 + 15
10a = 40
10a/10 = 40
a = 4
(one solution)
-2d + 4d - 4 = 5 + 3(d - 3)
2d - 4 = 5 + 3d - 9
2d - 3d = 5 - 9 + 4
- d = 0
d = 0
(one solution)
-11(n - 2) + 8(n + 1) = -3(n + 10)
-11n + 22 + 8n + 8 = -3n - 30
-11n + 8n + 3n = -30 - 22 - 8
0 = -60
(no solutions)
Give the name (monomial,
binomial, trinomial etc.) and
degree of the polynomial.
3x – 4
Answer:
degree- 1
name- binomial i think
PLEASE ANSWER ASAP FOR BRANLEST!!!!!!!!!!!!!!!
Solve,Check, show work:
a) 2x + 7 = 15
b) 3/4x + 8 = -12
A)11
B)0.1875
Step-by-step explanation:
2x+7=15
2x+7-7=15+7
2x=22
2x/2=22/2
3.4x+8=-12
3/4x+8-8=-12+8
3/4x=-4
3/4divided by 3/4=-4 divided by 3/4
x=0.1875
SOMEONE HELP PLEASE !!!
n^2 - m; use m = 7, and n = 8
Answer:
57
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightStep-by-step explanation:
Step 1: Define
n² - m
m = 7
n = 8
Step 2: Evaluate
Substitute: 8² - 7Exponents: 64 - 7Subtract: 57Write an equation of the line that passes through (−2,3) and (0,8).
Answer:
[tex]y = \frac{5}{2}x + 8[/tex]
Step-by-step explanation:
(-2 , 3 ) ; (0 , 8)
Slope = [tex]\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]
[tex]= \frac{8-3}{0-[-2]}\\\\=\frac{5}{0+2}\\\\=\frac{5}{2}[/tex]
m = 5/2 ; (0 , 8)
Equation: y -y1 = m(x -x1)
[tex]y - 8 =\frac{5}{2}(x - 0)\\\\y - 8 =\frac{5}{2}x\\y = \frac{5}{2}x + 8[/tex]
Is 8/18 closer to 0 1/2 or 1
Answer:
Closer to 1\2
Step-by-step explanation:
half of 18 is 9 and if you add 1 to 8 you get 9 so 8\18 is closer to 1\2
Hope it helps!
The area of a square field is 4,900 square meters. How long is one side
of the field?
O 70 meters
O 140 meters
O 560 meters
O 1225 meters
Answer:
1225
Step-by-step explanation:
divide by 4 because a square has 4 sides
A bag contains 15 blue marbles, 5 red marbles, and 11 green marbles. How many blue marbles must be removed from the 31 marbles in the bag so that the probability of randomly drawing a blue marble is 1/3?
1. 7
2. 12
3. 6
4. 9
5. 11
Answer:
7
Step-by-step explanation:
Hello? STOP RIGHT THERE.... Please help me with this.
Answer: THE ANSWER IS 2 :)
Step-by-step explanation:
3(q+2) Do distributive property
3q=6 Divide 3q to both side
3q = 3q
q= 2 This is the answer
Mia is making pickles. She uses 3/4 cup of salt when she has 9 pounds of cucumbers. How much salt will Mia use when she has 3 pounds of cucumbers?
Answer:
1/4 cup of salt. .........
The intensity of the sound, I watts/m², received from a loudspeaker is
inversely proportional to the square of the distance, d metres, from the
loudspeaker.
When d = 2,1 = 30
Work out the value of I when d = 10
PLEASE HELP
Answer:
The value of I for d = 10 will be 1.2 watts/m^3
Step-by-step explanation:
Let I be the intensity of sound and d be the distance
According to given statement, "The intensity of the sound, I watts/m², received from a loudspeaker is inversely proportional to the square of the distance, d meters, from the loudspeaker. "
[tex]I \propto \frac{1}{d^2}[/tex]
Removing the proportionality symbol
[tex]I = \frac{k}{d^2}[/tex]
Putting d = 2 and I = 30
[tex]30 = \frac{k}{(2)^2}\\30 = \frac{k}{4}\\k = 30*4\\k = 120[/tex]
Putting the value of k in the equation
[tex]I = \frac{120}{d^2}[/tex]
Putting d = 10
[tex]I = \frac{120}{(10)^2}\\I = \frac{120}{100}\\I = 1.2[/tex]
Hence,
The value of I for d = 10 will be 1.2 watts/m^3
How many one and a half L bottles does it take to fill a 16 L jug
Answer:
hmmm. i woud answer this question but i do not know how many ounces the bottles have? is this a real question or a joke,because i coud maybe help if you could be more specific.
Step-by-step explanation:
Plz help me it says show work to I’m confused
Answer:
1 = 7x=42 (divide both sides by 7)
7x/7 =42/7
x=6
2.=9y=81 (divide both sides by 9)
9y/9y= 81/9
y=9
3.= –40=8a (divide both sides by 8)
–40/8= 8a/8
a=5
4.=y divided 11=12
y=11 multiplied by 12
y= 132
5.= –15y=135( divide both sides by –15)
–15y/–15=135/–15
y= –9
6.= 17p=51(divide both sides by 17)
17p/17=51/17
p=3
7.= –65= –13w (divide both sides by 13)
–65/–13=–13w/–13
w=5
8.=y/–2=4
(cross multiply)
y/–2=4/1
4 multiplied by –2= –8
therefore:y= –8
Step-by-step explanation:
I HOPE THIS HELPED :)
Answer:
a) 7x=42
Step-by-step explanation
use the co-efficient of the unknown to divide both sides
7x=42
7x/7=42/7
x=6
9y=81
9y/9=81/9
y=9
-40=8a
is the same thing as
-8a=40
-8a/-8=40/-8
a=-5
y+11=12
y=12-11
y=1
-15y=135
-15y/-15=135/-15
y=9
17p=51
17p/17=51/17
p=3
-65=-13w
13w=65
13w/13=65/13
w=5
y/-2= 4
cross multiply y=-8
A diagram of a swimming pool is shown.
3.5 m
What is the approximate circumference of the swimming pool? Use 3. 14 as an estimate for .
O A. 7m
OB. 11 m
OC. 22 m
OD. 28 m
Answer:
Yeah the answer is 22
Step-by-step explanation:
Good job! :)
Solve the inequality
Answer:
x ≥ -64
Step-by-step explanation:
"3/4ths of a number is greater than or equal to -48"
3/4 of a number ≥ -48
Let's say the number is x.
(3/4)x ≥ -48
Multiply both sides by 4.
3x ≥ -192
Divide both sides by 3.
x ≥ -64
The variables y and x have a proportional relationship, and y = 9 when x = 2. What is the value of y when x = 3? Enter your answer as a decimal in the box.
Which three side lengths best describe the triangle in the diagram?
O NG
1
2
3
0 1 2
3 4
5 6 7 8 9 10 11 12 13
LLLLLLL
12 13
5
8 9 10 11
4
O NCS 2 3
2
O NOS
O A. 5 cm, 12 cm, and 13 cm
OB. 5 cm, 10 cm, and 13 cm
O C. 4 cm, 8 cm, and 12 cm
OD. 7 cm, 12 cm, and 13 cm
Step-by-step explanation:
Option A best describes the diagram given above.