In part (a) of Question 1, we are asked to determine whether certain matrices are orthogonal or not, given that matrices A and B are orthogonal. We need to justify our answers. Specifically, we need to determine if AT (transpose of A), B-¹ (inverse of B), and B-¹AB are orthogonal matrices.
In part (b), we are given a transformation f from R² to R³ and asked to show that it is a linear transformation. Additionally, we need to find a matrix representation of this transformation.
(a) To determine if AT, B-¹, and B-¹AB are orthogonal matrices, we need to recall the properties of orthogonal matrices. An orthogonal matrix is a square matrix whose transpose is equal to its inverse.
(i) For AT, since A is orthogonal, we know that AAT = I, where I is the identity matrix. Taking the transpose of both sides, we have (AAT)T = IT, which simplifies to ATA = I. Therefore, AT is also orthogonal.
(ii) For B-¹, if B is orthogonal, then BB-¹ = I. Taking the transpose of both sides, we have (BB-¹)T = IT, which simplifies to (B-¹)TBT = I. Therefore, B-¹ is orthogonal.
(iii) For B-¹AB, we can rewrite it as (B-¹A)B. Since A and B are orthogonal matrices, B-¹A is also orthogonal. Multiplying an orthogonal matrix (B-¹A) by another orthogonal matrix (B) results in another orthogonal matrix. Therefore, B-¹AB is orthogonal.
(b) To show that f is a linear transformation, we need to verify two properties: additivity and homogeneity.
Additivity: f(u + v) = f(u) + f(v) for any vectors u and v in R².
Let u = (x₁, y₁) and v = (x₂, y₂). Then f(u) = (x₁ + 2y₁, x₁ - y₁, -2x₁ + 3y₁) and f(v) = (x₂ + 2y₂, x₂ - y₂, -2x₂ + 3y₂). Adding these two vectors, we get f(u + v) = (x₁ + x₂ + 2(y₁ + y₂), x₁ + x₂ - (y₁ + y₂), -2(x₁ + x₂) + 3(y₁ + y₂)). By comparing f(u + v) with f(u) + f(v), we see that they are equal. Hence, f satisfies additivity.
Homogeneity: f(cu) = cf(u) for any scalar c and vector u in R².
Let u = (x, y). Then f(u) = (x + 2y, x - y, -2x + 3y). Multiplying this vector by scalar c, we get f(cu) = (cx + 2cy, cx - cy, -2cx + 3cy). By comparing f(cu) with cf(u), we see that they are equal. Hence, f satisfies homogeneity.
To find the matrix representation of f, we can write the coordinates of f(u) as a linear combination of the standard basis vectors in R³:
f(u) = x(1, 1, -2) + y(2, -1, 3)
The matrix representation of f is then:
[1 2]
[1 -1]
[-2 3]
This matrix can be used to perform vector operations on u directly.
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a cylindrical brass pipe is 600 cm in length. its outside diameter is 10 cm, and its inside radius is 4 cm. what is the total surface area of this pipe (including the internal surface area of this pipe)?
The total surface area of the cylindrical brass pipe, including the internal surface area, is [tex]10841π cm^2[/tex], where π is approximately 3.14159.
To calculate the total surface area of the cylindrical brass pipe, we need to consider both the external and internal surface areas.
The external surface area of a cylinder is given by the formula:
A_external =[tex]2πrh + πr^2[/tex]
Where:
r is the radius of the outer surface (half of the outside diameter), which is 10 cm / 2 = 5 cm
h is the height of the cylinder, which is given as 600 cm
Plugging in the values:
A_external = [tex]2π(5 cm)(600 cm) + π(5 cm)^2[/tex]
= [tex]2π(3000 cm^2) + π(25 cm^2)[/tex]
= [tex]6000π cm^2 + 25π cm^2[/tex]
= [tex]6025π cm^2[/tex]
The internal surface area of a cylinder is given by the formula:
A_internal = [tex]2πrh + πr^2[/tex]
Where:
r is the radius of the inner surface (inside radius), which is given as 4 cm
h is the height of the cylinder, which is given as 600 cm
Plugging in the values:
A_internal = [tex]2π(4 cm)(600 cm) + π(4 cm)^2[/tex]
= [tex]2π(2400 cm^2) + π(16 cm^2)[/tex]
= [tex]4800π cm^2 + 16π cm^2[/tex]
= [tex]4816π cm^2[/tex]
To find the total surface area, we add the external and internal surface areas:
Total surface area = A_external + A_internal
= [tex]6025π cm^2 + 4816π cm^2[/tex]
= [tex]10841π cm^2[/tex]
Therefore, the total surface area of the cylindrical brass pipe, including the internal surface area, is [tex]10841π cm^2[/tex], where π is approximately 3.14159.
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Texas predicted that the population would be about 17.4 million in 2018 and then increase by 0.22 million per year until 2020. Which linear equations models this situation?
A. y=17.4x +0.22
B. y= -0.22x + 17.4
C. y= 0.22x + 17.4
D. y= -17.4x + 0.22
Answer:
C. y = 0.22x + 17.4
Step-by-step explanation:
0.22x represents how much it's growing by. 17.4 represents the initial value.
y represents the final value.
Answer:
C. y= 0.22x + 17.4
Step-by-step explanation:
y= mx + b
0.22 million per year = m
17.4 = b
# of years = x
Total population = y
y= 0.22x + 17.4
If (-3,5) and (3,-5) are two points that lie on function f, then is f even, odd, or neither?
f(-3) ≠ f(3) and f(-3) ≠ -f(3), which means that the function f does not satisfy the properties required for it to be even or odd.
To determine whether the function f is even, odd, or neither, we need to evaluate the symmetry of the function with respect to the y-axis and the origin.
For a function to be even, it must satisfy the property f(x) = f(-x) for all values of x in the function's domain. This means that if we substitute -x for x in the function, we should obtain the same output as when we evaluate the function at x.
For a function to be odd, it must satisfy the property f(x) = -f(-x) for all values of x in the function's domain. This means that if we substitute -x for x in the function, we should obtain the negative of the output we obtain when we evaluate the function at x.
Let's evaluate the function f using the given points (-3,5) and (3,-5):
For the point (-3,5):
f(-3) = 5
For the point (3,-5):
f(3) = -5
We can see that f(-3) ≠ f(3) and f(-3) ≠ -f(3), which means that the function f does not satisfy the properties required for it to be even or odd.
We can conclude that the function f is neither even nor odd.
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write each equation in vertex form. then identify the vertex, axis of symmetry and direction of opening. y=x^2+8x+\:18 , y=-\:x^2\:\:12x\:-\:36 and y=2x^2\:+\:12x\:+\:13
The equation in vertex form is y = 2(x+3)² + 1. The vertex is (-3,1), the axis of symmetry is x = -3, and the direction of opening is upwards.
The vertex form of the equation is y = a(x-h)^2 + k, where (h, k) is the vertex. The axis of symmetry is x = h, and the direction of opening is determined by the value of a.
Using this formula, let us write each equation in vertex form and then identify the vertex, axis of symmetry, and direction of opening.
1. y = x² + 8x + 18
To write this equation in vertex form, we need to complete the square. y = x² + 8x + 18 is equivalent to y = (x+4)² - 2. Therefore, the equation in vertex form is y = (x+4)² - 2.
The vertex is (-4,-2), the axis of symmetry is x = -4, and the direction of opening is upwards.2
. y = -x² - 12x - 36To write this equation in vertex form, we need to complete the square. y = -x² - 12x - 36 is equivalent to y = -(x+6)² - 12. Therefore, the equation in vertex form is y = -(x+6)² - 12.
The vertex is (-6,-12), the axis of symmetry is x = -6, and the direction of opening is downwards.3. y = 2x² + 12x + 13To write this equation in vertex form, we need to complete the square. y = 2x² + 12x + 13 is equivalent to y = 2(x+3)² + 1.
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PLEASE HELP IM CONFUSED! Find what the measurements of angles a-c are and state the reason
Answer:
<a = 90° (180-90 supplementary angle)
<b = 90° (180-90 supplementary angle)
<d = 48° (180-132 supplementary angle)
<e = 132° (opposite angle)
<c = 42° (interior angles of a triangle equal 180. 180-48-90=42)
Step-by-step explanation:
<a = 90° (180-90 supplementary angle)
<b = 90° (180-90 supplementary angle)
<d = 48° (180-132 supplementary angle)
<e = 132° (opposite angle)
<c = 42° (interior angles of a triangle equal 180. 180-48-90=42)
18×2×5 equal 18×2 times
Answer:
no
Step-by-step explanation:
The empire state building stands at a height of 1,454 feet. Given the choice between feet feet per second or miles per hour, which rate of measure is more reasonable to describe the speed of a penny when dropped off the top of the building? Justify your reasoning using mathematical laguage
Answer:
miles per hour
Step-by-step explanation:
speed can be define as the rate of distance covered in a given time.
i.e speed = [tex]\frac{distance}{time}[/tex]
Given that the height (distance) of the empire state building = 1454 feet.
a. For feet per second;
distance = 1454 feet
time = 1 second
speed = [tex]\frac{1454}{1}[/tex]
= 1454 feet/second
b. For miles per hour,
1 feet = 0.000189394 mile
Thus, 1454 feet = 0.2753788 mile
distance = 0.2753788 mile
time = 1 hour
speed = [tex]\frac{0.2753788}{1}[/tex]
= 0.2753788 miles/hour
Comparing the two units of measurement, miles per hour is more reasonable. This is because the penny can not move at that very high speed to hit the ground within 1 second when dropped.
A garden table and a bench cost $781 combined. The garden table costs $69 less than the bench. What is the cost of the bench?
Answer:
$425
Step-by-step explanation:
x = bench
x - 69 = garden table
x + (x - 69)=781
x + x - 69 =781
2x - 69 = 781
2x -69 +69 = 781 + 69
2x = 850
2x/2 = 850/2
x = 425
Bench = 425
Answer: £425?
Step-by-step explanation:
x = bench
x - 69 = garden table
x + (x -69) = 781
2x - 69=781
+ 69 both sides
2x = 850
x = 425
nine less than 15 times a number is 6 write as a equation
Let the number = x
You have 15 times the number so that is 15x
9 less would be subtracting 9:
15x -9
Is 6 means the equation equals 6:
The equation is: 15x-9 = 6
10 Murpy has $30, but he spends $2 every day at lunch. use the table to figure out how much money he will have left each day.
Please respond quickly
Answer:
(0,30)
(1,28)
(2,26)
(5,20)
Step-by-step explanation:
If you multiply 2 times the x
ex. 2 x 5 = 10
Then subtract that from the 30 to get y
30-10 = 20 = y
Solve
(5,20)
Answer:
one day and he has 28 dollars
Step-by-step explanation:
24. 075 21.048 24.008 21 .605 how do i answer this from greatest to least
Answer:
24.075, 24.008, 21.605, 21.048
A person 6 feet tall casts a shadow 4 feet long. At the same time, a nearby tower casts a shadow 32 feet long. How many feet are in the height of the tower?
Answer:
48
Step-by-step explanation:
First 4 is 2/3 of 6 feet
Thus 32 is 2/3 of x
How do we find X
32 divided by 2 is 16
16 plus 32 = 48
How many factors does 144 have?
08
O 12
O 15
06
Answer:
15
Step-by-step explanation:
1, 4, 2, 3, 6, 8, 16, 9, 12, 18, 24, 36, 48, 72, 144.
Answer:
15
Step-by-step explanation:
1 , 2 , 3 , 4 , 6 , 8 , 9 , 12 , 16 , 18 , 36 , 48 , 72 , 144
neptune is an average distance of 4.5×109km from the sun. estimate the length
We are to estimate the length of the Neptune which has an average distance of 4.5 × 10^9 km from the sun. The length of Neptune can be estimated using the circumference of its orbit.
We are given that Neptune is an average distance of 4.5 × 10^9 km from the sun. To estimate the length of Neptune, we can use the circumference of its orbit. The circumference of Neptune's orbit can be estimated using the formula;
Circumference (C) = 2πr
Where C is the circumference, r is the radius and π is a mathematical constant equal to 3.14.
Since we are given the average distance of Neptune from the sun, we can estimate the length of Neptune using the above formula.Substituting the values given into the formula;
Circumference (C) = 2πr = 2 × 3.14 × 4.5 × 10^9 kmC = 28.26 × 10^9 km
Hence, the circumference of Neptune's orbit is approximately 28.26 × 10^9 km. Since the circumference of a circle is equal to 2πr, we can estimate the radius of the orbit by dividing the circumference by 2π;
r = C / 2π = (28.26 × 10^9 km) / (2 × 3.14) = 4.50 × 10^9 km
Hence, the radius of Neptune's orbit is approximately 4.50 × 10^9 km.
Since Neptune's orbit is almost circular, its length can be estimated using the formula;
Length = 2πrLength = 2 × 3.14 × 4.5 × 10^9 kmLength = 28.26 × 10^9 km
Hence, the length of Neptune is approximately 28.26 × 10^9 km.
Therefore, we can estimate the length of Neptune using the circumference of its orbit which is approximately 28.26 × 10^9 km.
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-2x + 8 = 18 please help. What does x equal
Answer:
x = -5
Step-by-step explanation:
Step 1: Write equation
-2x + 8 = 18
Step 2: Solve for x
Subtract 8 on both sides: -2x = 10Divide both sides by -2: x = -5Step 3: Check
Plug in x to verify it's a solution.
-2(-5) + 8 = 18
10 + 8 = 18
18 = 18
Answer:
x = -5
Step-by-step explanation:
-2x + 8 = 18
Subtract 8 from each side
-2x + 8-8 = 18-8
-2x = 10
Divide each side by -2
-2x/-2 = 10/-2
x = -5
Triangles T S R and Q R S overlap at point P. Angle S T R is 66 degrees and angle S Q R is 66 degrees. The lengths of T S and Q R are both 5 inches. Can ΔTSR and ΔQRS be proven congruent by SAS?
Answer:
D- no, because not enough is information given to prove the triangles congruent by SAS
Step-by-step explanation:
^^I gotchu kuh
Answer:
the answer is d
Step-by-step explanation:
Solve for v. 23= -81-7+2v
Answer:
v = 55.5
Step-by-step explanation:
23 = -81 - 7 + 2v
23 = -88 + 2v
23 + 88 = -88 + 88 + 2v
111 = 0 + 2v
111 = 2v
111 ÷ 2 = 2v ÷ 2
55.5 = 1v
55.5 = v
Answer:
v = 55.5
Step-by-step explanation:
23 = -81 - 7 + 2v
1. Simplify by combining like terms
23 = -88 + 2v
2. Isolate the variable by adding 88 to both sides
23 + 88 = -88+ 88 +2v
23 + 88 = 2v
3. Combine like terms
111 = 2v
4. Solve for v by dividing both sides by 2
111/2 = 2v/2
55.5 = v
−12.405 as a mixed number in simplest form.
Answer:
-12 81/200
Step-by-step explanation:
That is the simplest it will go.
Select the true statements regarding the disk and washer methods of integration. a. The cross sections of a solid of revolution can be either washers or disks. b. Examples of solids of revolution include the sphere, cone, and cylinder.
c. The disk method is appropriate in finding the volume of a solid obtained through the rotation of the region bounded by a function with respect to y about the x-axis. d. The volume of a solid of revolution obtained through rotation about the horizontal line y = c involves integration with respect to x. e. The washer method is appropriate in finding the volume of a solid obtained through the rotation of the region bounded by a function y = f(x) about the x-axis.
The following are the true statements regarding the disk and washer methods of integration:
a. The cross sections of a solid of revolution can be either washers or disks.
b. Examples of solids of revolution include the sphere, cone, and cylinder.
e. The washer method is appropriate in finding the volume of a solid obtained through the rotation of the region bounded by a function y = f(x) about the x-axis.
Disk method of integration: This method is appropriate in Calculus for finding the volume of a solid obtained by revolving a region bound by a function, y = f (x) about the horizontal axis x. The method takes a slice of the solid perpendicular to the x-axis. The cross-section of the solid obtained by slicing it perpendicular to the x-axis will be a circular disk of radius f (x).
Washer method of integration: The washer method is appropriate for finding the volume of a solid obtained by revolving a region bound by a function, y = f (x), about a horizontal axis. The method involves taking a slice of the solid perpendicular to the x-axis. The cross-section of the solid obtained by slicing it perpendicular to the x-axis will be a circular washer with an inner radius of g (x) and an outer radius of f (x).
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What is the conjugate of 4+i?
Answer:
4-i
Step-by-step explanation:
The complex conjugate means take the imaginary term and negate it
4+i becomes 4-i
4-i is the complex conjugate
How do you get the answer to this question?
Answer:
2 small notebooks and 4 large notebooks
Step-by-step explanation:
$8 × 2 = $16
$10 × 4 = $40
2 + 4 = 6
$40 + $16 = $56
Hope that helps.
Spencer built a tower of blocks that was 3.2 centimeters high. He used 2 identical blocks to build the tower. What was the height of each of the blocks?
Answer:
1.8
Step-by-step explanation:
Hope this helps
A rectangular mural measures 1.3 feet by 4.8 feet. harmony creates a new mural that is 0.6 feet longer. what is the perimeter of harmony`s new mural?
The Perimeter of Harmony's new mural is 13.4 feet.
To determine the perimeter of Harmony's new mural, we need to use the formula for the perimeter of a rectangle, which is:P = 2l + 2w
where P represents the perimeter, l represents the length, and w represents the width of the rectangle. The units of measurement are in feet.
Given that the rectangular mural measures 1.3 feet by 4.8 feet, we can plug these values into the formula to find the perimeter of the original mural:P = 2(1.3) + 2(4.8)P = 2.6 + 9.6P = 12.2
Therefore, the perimeter of the original mural is 12.2 feet.
Now, Harmony creates a new mural that is 0.6 feet longer than the original mural. This means that the length of the new mural is 1.3 + 0.6 = 1.9 feet.
To find the perimeter of the new mural, we need to plug this new length into the same formula:P = 2l + 2wP = 2(1.9) + 2(4.8)P = 3.8 + 9.6P = 13.4
Therefore, the perimeter of Harmony's new mural is 13.4 feet.
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Which of the following combinations would result in multiple triangles? SHOW WORK
A: 45 degrees, 82 degrees, 53 degrees
B: 5 degrees, 122 degrees, 61 degrees
C: 2 cm, 8 cm, 10 cm
D: 10 in, 20 in, 75 degrees
Multiple triangles would be formed by the combination of 2 cm, 8 cm, and 10 cm.
To have a triangle, three sides of a polygon must be present. Additionally, each side must be less than the total of the other two sides added together to form a triangle.
Consider the following choices as a result:
A: 45 degrees, 82 degrees, 53 degrees.
We don't know the size of the triangle's sides, thus we can't say if it's going to create multiple triangles
.B: 5 degrees, 122 degrees, 61 degrees
This set of degrees won't result in a triangle because the angles add up to 188 degrees, which is greater than 180 degrees, the sum of the interior angles of a triangle.
C: 2 cm, 8 cm, 10 cm
The sum of 2 cm + 8 cm = 10 cm.
As a result, it is impossible for a triangle to be formed since 2 + 8 < 10.D: 10 in, 20 in, 75 degreesTwo sides of the triangle are known, however the angle between them is unknown, therefore it is impossible to tell if multiple triangles will form.
Option C is the only choice that would result in multiple triangles. The side lengths of a triangle are 2 cm, 8 cm, and 10 cm. It is impossible to form a triangle with 2 cm and 8 cm because they are too short to form a triangle, however if you switch the values and compare 8 cm and 10 cm, a triangle is formed.
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What is the slope of the line passing through the points (1, - 5) and (4, 1)?
Answer:
2
Step-by-step explanation:
We can use the two point to find the slope
m = (y2-y1)/(x2-x1)
= ( 1- -5)/(4-1)
= (1+5)/(4-1)
= 6/3
=2
One uranium atom has a diameter of 3.5x10^-8 centimeters. What is the sum of 10,000 uranium atoms'
diameters, in centimeters, written in standard notation?
Answer:
3.5 * 10^-4 centimeters
Step-by-step explanation:
Here, we have one uranium atom having a diameter of 3.5 * 10^-8 centimeters.
We now want to find the sum of 10,000 uranium atoms diameters
Mathematically;
That would be ;
3.5 * 10^-8 * 10,000
But 10,000 = 10^4
Thus, we have
3.5 * 10^-8 * 10^4
= 3.5 * 10^-4 centimeters
I NEED HELP ASAP IT'S THE LAST QUESTION THAT I NEED TO GET 100 PERCENT!!!!
Answer:
The exact form would be 2/25but in decimal form it is 0.08
Step-by-step explanation:
GrantWhich of the following is the graph of the inequality y < -2x + 3?
The graph that represents the linear inequality y < -2x + 3 is graph B
What is graph of linear inequality?The set of solutions to a linear inequality are represented graphically on a coordinate plane. It makes the area or areas that satisfy the inequality easier to visualize.
The linear equation that results from changing the inequality sign to an equal sign is the equation that is commonly graphed first when a linear inequality has to be represented. The coordinate plane is split into two regions by this linear equation: one zone where the linear equation holds true, and another region where it does not.
Next, we determine which side of the line represents the solution set for the inequality. This is determined by the inequality sign (<, >, ≤, ≥) and the given inequality's conditions.
To solve this problem, we can simply plot this using a graphing calculator ;
Kindly find the attached graph of the linear inequality y < -2x + 3 below.
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Pablo runs 6 miles in 50 minutes. At the same rate, how many miles would he run in 35 minutes?
Answer:
4.2 miles
Step-by-step explanation:
Answer:
4.2 miles
Step-by-step explanation:
6/50=0.12
0.12x35=4.2
Find the unit rate. _____ (number) per ounce
$1.29 for 20.5 oz.
Answer:
15.8914728682 is the long answer, but simplified, it's 15.9
Step-by-step explanation:
Just do 20.5 ÷ 1.29
