Find a basis for the eigenspace corresponding to each listed eigenvalue. a = [7 9; -2 -2], = 1,4,
Answers
The basis for the eigenspace corresponding to the listed eigenvalue is λ₁ = 1: Basis eigenvector v₁ = [-3/2; 1] and λ₂ = 4: Basis eigenvector v₂ = [1; -1/3].
To find the eigenspaces corresponding to the eigenvalues of a given matrix, we need to solve the equation (A - λI) v = 0, where A is the matrix, λ is the eigenvalue, I is the identity matrix, and v is the eigenvector.
Given matrix A:
A = [7 9; -2 -2]
Eigenvalues:
λ₁ = 1
λ₂ = 4
For each eigenvalue, we need to find the eigenvectors.
For λ₁ = 1:(A - λ₁I) v₁ = 0
Substituting the values:
([7 9; -2 -2] - 1 * [1 0; 0 1]) v₁ = 0
[6 9; -2 -3] v₁ = 0
Solving the system of equations:
6v₁ + 9v₂ = 0
-2v₁ - 3v₂ = 0
We can choose a value for v₂, say v₂ = 1:
6v₁ + 9(1) = 0
-2v₁ - 3(1) = 0
Simplifying the equations:
6v₁ + 9 = 0
-2v₁ - 3 = 0
Solving these equations, we find v₁ = -3/2.
Therefore, for λ₁ = 1, the eigenvector v₁ is [-3/2; 1].
For λ₂ = 4:(A - λ₂I) v₂ = 0
Substituting the values:
([7 9; -2 -2] - 4 * [1 0; 0 1]) v₂ = 0
[3 9; -2 -6] v₂ = 0
Solving the system of equations:
3v₁ + 9v₂ = 0
-2v₁ - 6v₂ = 0
We can choose a value for v₁, say v₁ = 1:
3(1) + 9v₂ = 0
-2(1) - 6v₂ = 0
Simplifying the equations:
3 + 9v₂ = 0
-2 - 6v₂ = 0
Solving these equations, we find v₂ = -1/3.
Therefore, for λ₂ = 4, the eigenvector v₂ is [1; -1/3].
In summary, the basis for the eigenspace corresponding to each eigenvalue is:
For eigenvalue λ₁ = 1: Basis eigenvector v₁ = [-3/2; 1]
For eigenvalue λ₂ = 4: Basis eigenvector v₂ = [1; -1/3]
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Related Questions
what are four methods of acquiring knowledge
Answers
The four methods of acquiring knowledge are observation, experience, reasoning, and education.
Observation involves gathering information through the senses and paying attention to what is happening in the environment. Experience involves learning through personal encounters, such as practical work or life events. Reasoning involves using logic and critical thinking to draw conclusions and make decisions.
Education involves acquiring knowledge through formal instruction and learning, such as in a school setting or through reading and studying. All four methods are important and can be used in combination to acquire a deep and well-rounded understanding of a subject.
Observation and experience provide hands-on and practical knowledge, while reasoning and education provide theoretical and abstract knowledge. By using a combination of these methods, individuals can broaden their understanding and knowledge of the world around them.
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A 0. 001 kg bullet is fired with a velocity of 800 m/s into a soft wood of mass 1 kg, resting on a smooth surface. What is the impulse of the bullet?
Answers
The impulse of a bullet is the product of the force and the time for which it is applied to the object. The impulse of a bullet can be calculated using the equation.The impulse of the bullet is 0.8 Ns.
Impulse = Force x Time.The given question states that a 0.001 kg bullet is fired with a velocity of 800 m/s into a soft wood of mass 1 kg, resting on a smooth surface. To calculate the impulse of the bullet, we need to determine the force applied to the wood by the bullet during the time of impact.First, let us find the initial momentum of the bullet, which is:Initial momentum = mass x velocity = 0.001 kg x 800 m/s = 0.8 kg m/sWhen the bullet hits the wood, it comes to rest. Therefore, the final velocity of the bullet is 0 m/s. To find the force exerted on the wood, we can use the law of conservation of momentum.
The total momentum of the system before the collision is equal to the total momentum of the system after the collision, which is:Initial momentum of bullet = Final momentum of bullet + Momentum of wood0.8 kg m/s = 0 + (mass of wood) x (final velocity of wood)We know that the mass of wood is 1 kg and the final velocity of wood is unknown. Solving for final velocity of wood gives:Final velocity of wood = 0.8 m/sSo, the change in velocity of the wood is:Δv = final velocity - initial velocity = 0.8 m/s - 0 m/s = 0.8 m/sThe time of impact can be calculated using the equation:Time = Δv / aWhere, a is the acceleration of the wood, which can be found using the equation:F = maF = mΔv / tTherefore,a = F / m = (mΔv / t) / m = Δv / tSubstituting the given values, we get:a = 0.8 m/s / 0.001 s = 800 m/s²The time of impact is:Time = Δv / a = 0.8 m/s / 800 m/s² = 0.001 sTherefore, the force exerted on the wood by the bullet is:F = ma = 1 kg x 800 m/s² = 800 NThe impulse of the bullet can be calculated using the equation:Impulse = Force x Time= 800 N x 0.001 s= 0.8 NsAnswer: The impulse of the bullet is 0.8 Ns.
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If a solenoid were suspended by a string so that it could rotate freely, could it be used as a compass when it carried a direct current? Could it also be used if the current were alternating in direction?
Answers
Yes, a solenoid suspended by a string can be used as a compass when it carries a direct current. This is because the solenoid creates a magnetic field when current flows through it.
The magnetic field aligns with the Earth's magnetic field, causing the solenoid to point in the direction of magnetic north. However, if the current were alternating in direction, the solenoid would constantly change its orientation and would not be reliable as a compass.
This magnetic field interacts with Earth's magnetic field, causing the solenoid to align itself with the magnetic north and south poles, acting as a compass.
Consequently, a solenoid with an alternating current would not function effectively as a compass.
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If you ran at an average speed of 8 m/s, how long would it take you to run 200 meters?
Answers
200/8 = 25
6.) A car traveling 30 m/s north has a momentum of 60,000 kg m/s. What is the mass of the car?
please help!! answer and equation would b great!!
Answers
Answer:
The answer is 2000 kgExplanation:
The mass of the car can be found by using the formula
\(m = \frac{p}{v} \\ \)
p is the momentum
v is the velocity
From the question we have
\(m = \frac{60000}{30} = \frac{6000}{3} \\ \)
We have the final answer as
2000 kgHope this helps you
if a person weighs 140 lb on earth their weight in newtons is
Answers
Answer:
623.6N
Explanation:
There are 2.2 pounds in one kilogram, so you must divide 140 lb by 2.2 to convert the units to kilograms.
140/2.2 = 63.6 kg
The gravitational force also known as weight would equal mass times acceleration due to gravity.
Fg = (63.6 kg)(9.8 m/s^2) = 623.6N
Which phrase best describes the outer planets?
Answers
Answer:
Are Gaseous with low density. (D)
Explanation:
I took the test on Edge and got it right
Hope this helps :)
Answer:
d
Explanation:
why biologists often use the term energy source as a synonym for electron donor.
Answers
Biologists often use the term "energy source" as a synonym for "electron donor" because the transfer of electrons is a key component of energy generation in many biological systems.
In biological systems, energy is typically generated through a series of chemical reactions called cellular respiration. In this process, electrons are transferred between molecules, releasing energy that is used to power cellular processes. The molecule that donates electrons is known as the electron donor, and the molecule that accepts the electrons is known as the electron acceptor.
Since the transfer of electrons is the primary mechanism by which energy is generated in these systems, biologists often use the term "energy source" to refer to the electron donor. For example, in photosynthesis, the energy source is the electron donor (usually water), which donates electrons to the photosynthetic electron transport chain, generating energy that is used to create ATP and reduce carbon dioxide to produce organic molecules.
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An axon of a human nerve cell is 5 x 10-6m in radius and 0.5-mm long. If the resistivity of the cytoplasm (inside the axon) is 1.6 x 107 Ω.m; Calculate the resistance along the axon, Raxial.
Raxial = ----- Ω
Answers
The axial resistance of an axon is calculated using the formula R = ρL/A, where ρ is the resistivity, L is the length, and A is the cross-sectional area. In this case, the axial resistance is 11.28 MΩ.
The resistance along the axon is calculated using the following formula:
R = ρL/A
where:
R is the resistance in ohms
ρ is the resistivity in ohms per meter
L is the length in meters
A is the cross-sectional area in meters squared
In this case, we have:
ρ = 1.6 x 107 Ω.m
L = 0.5 mm = 0.0005 m
A = πr² = π(5 x 10-6)² = 7.854 x 10-13 m²
Therefore, the resistance is:
R = ρL/A = (1.6 x 107 Ω.m)(0.0005 m) / (7.854 x 10-13 m²) = 11.28 MΩ
Therefore, the axial resistance of the axon is 11.28 MΩ.
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Where does solar energy wastes go
Answers
Answer:
Space
Explanation:
Solar energy is energy from the Sun. This energy is in form of radiation heat and light. When solar energy reaches a surface it bounces off. This is because they are radiation waves.
How many valence electrons are in an atom of fluorine?
two
seven
nine
seventeen
Answers
Is water a element or compound or a mixture
Answers
Answer:
Water is a compound because it is made up of water molecules. There is no such thing as water atoms. Water molecules are made of hydrogen and oxygen atoms, in the definite proportion of two hydrogens for one oxygen.
How many windows should the real tree house have
Answers
Answer:
88
Explanation:
Answer:
2
Explanation:
88 was wrong and the answer then said 2
at room temperature, what fraction of the nitrogen molecules in the air are moving at less than 300 m/s?
Answers
The fraction of nitrogen molecules in the air that are moving at less than 300 m/s is likely to be very high, since this is well below the average speed of nitrogen molecules at room temperature. However, the exact fraction will depend on the specific temperature and pressure conditions.
At room temperature, the majority of nitrogen molecules in the air move at speeds less than 300 m/s. The average speed of nitrogen molecules in the air is around 500 m/s, but the speed distribution follows a bell-shaped curve, with a small fraction of molecules moving much faster and a small fraction moving much slower than the average.
The distribution of molecular speeds is determined by the Maxwell-Boltzmann distribution, which describes how the speeds of gas molecules are related to temperature. The distribution shows that at any given temperature, only a small fraction of molecules have speeds greater than a certain value.
For example, at room temperature (around 25°C or 298 K), only about 2.5% of nitrogen molecules in the air have speeds greater than 500 m/s, while the vast majority (over 97%) have speeds less than this value. Even fewer molecules (less than 0.1%) have speeds greater than 1000 m/s, which is much faster than the speed of sound in air.
Overall, the fraction of nitrogen molecules in the air that are moving at less than 300 m/s is likely to be very high, since this is well below the average speed of nitrogen molecules at room temperature. However, the exact fraction will depend on the specific temperature and pressure conditions.
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Electrolytes are compounds that conduct electricity in aqueous solutions. For a compound to be considered an electrolyte, what must happen when it dissolves in water?
Answers
When a compound dissolves in water to become an electrolyte, it must dissociate or break apart into ions.
These ions are then able to carry an electrical charge and conduct electricity in the aqueous solution. The greater the degree of dissociation, the stronger the electrolyte will be.
For example, when sodium chloride (NaCl) dissolves in water, it dissociates into sodium ions (Na+) and chloride ions (Cl-). These ions are then able to conduct electricity in the solution.
In contrast, a compound that does not dissociate into ions when dissolved in water is considered a nonelectrolyte and will not conduct electricity.
In summary, for a compound to be considered an electrolyte, it must dissociate into ions when dissolved in water, allowing it to conduct electricity in the aqueous solution.
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The engine starter and a headlight of a car are connected in parallel to the 12.0-V car battery. In this situation, the headlight operates at 38 W and the engine starter operates at 2.40 kW. If the headlight and starter were then rewired to be in series with each other, what total power would they consume when connected to the 12.0-V battery
Answers
Answer:
The total power they will consume in series is approximately 2.257 W
Explanation:
The connection arrangement of the headlight and the engine starter = Parallel to the battery
The voltage of the battery, V = 12.0 V
The power at which the headlight operates in parallel, \(P_{headlight}\) = 38 W
The power at which the kick starter operates in parallel, \(P_{kick \ starter}\) = 2.40 kW
We have;
P = V²/R
Where;
R = The resistance
V = The voltage = 12 V (The voltage is the same in parallel circuit)
For the headlight, we have;
R₁ = V²/\(P_{headlight}\) = 12²/38 = 72/19
R₁ = 72/19 Ω
For the kick starter, we have;
R₂ = V²/\(P_{kick \ starter}\) = 12²/2.4 = 60
R₂ = 60 Ω
When the headlight and kick starter are rewired to be in series, we have;
Total resistance, R = R₁ + R₂
Therefore;
R = ((72/19) + 60) Ω = (1212/19) Ω
The current flowing, I = V/R
∴ I = 12 V/(1212/19) Ω = (19/101) A
We note that power, P = I²R
In the series connection, we have;
\(P_{headlight}\) = I² × R₁
∴ \(P_{headlight}\) = ((19/101) A)² × 72/19 Ω = 1368/10201 W ≈ 0.134 W
The power at which the headlight operates in series, \(P_{headlight, S}\) ≈ 0.134 W
\(P_{kick \ starter}\) = ((19/101) A)² × 60 Ω = 21660/10201 W ≈ 2.123 W
The power at which the kick starter operates in series, \(P_{kick \ starter, S}\) ≈ 2.123 W
The total power they will consume, \(P_{Total}\) = \(P_{headlight, S}\) + \(P_{kick \ starter, S}\)
Therefore;
\(P_{Total}\) ≈ 0.134 W + 2.123 W = 2.257 W
The resistance of resistor is greater for:
a.Short and thin resistor
b.Short and thick resistor
c.Long and thin resistor
d.None
Answers
Answer:
c: long and thin resistor.
Explanation:
The resistance of a resistor is given by:
R = ρ*L/A
where:
R = resistance
ρ = resistivity (depends on the material)
L = length of the material
A = cross-sectional area of the material
We can see that the length is on the numerator, which means that if we increase the length, then the resistance is increased.
We also can see that the cross-sectional area is on the denominator, then if we increase the area (for example, with a ticker resistor) the resistance decreases.
Then if we want to maximize the resistance, we need to have a long and thin resistor, so the correct answer is c.
In a novel from 1866 the author describes a spaceship that is blasted out of a cannon with a speed of about 1.20×104 m/s . The spaceship is approximately 274 m long, but part of it is packed with gunpowder, so it accelerates over a distance of only 211m.
What was the acceleration (in m/s2 ) experienced by the occupants of the spaceship during launch? (Though the novel’s author realized that the “travelers would … encounter a violent recoil,” he probably didn’t know that people generally lose consciousness if they experience accelerations greater than about 7g∼70m/s2. )
Answers
Answer:341232 m/s^2
Explanation: you are given velocity, 1.20x10^4
distance, 211m (the 274m don't matter here)
with this information, you use the 3rd kinematic equation, vf^2-vi^2=2a(xf-xi)
vf and vi can just combine to become V (same with x)
divide ay on both sides to make a stay by itself
you are left with a=V^2/2x
plugin, 1.20x10^4/2x211
put it through a calculator and you get 341232 m/s^2
and I'm assuming you are doing MasterPhysics since I just did mine there so the answer for you is either 340000 m/s^2 or 3.4x10^5 m/s^2
HELP HELP ILL PUT BRAINLIEST TO WHO EVER DOES THIS CORRECT!!
Answers
The scale on the horizontal axis is 8 s per division and on the vertical axis 9 m per division. What is the time represented by the third tic mark on the horizontal axis?
Answer in units of s. What is the position represented by the fourth tic mark on the vertical axis? Answer in units of m. What is the position when t = 24 s? Answer in units of m.
Answers
Answer:
made 72
Explanation:
X axis says any of its values should be divison by 9. So you would multiply 8and 9 which gets you 72.
If your face is 62 cm in front of a plane mirror, where is the image of your face located?
62 cm in back of the mirror
62 cm below the mirror
62 cm in front of the mirror
31 cm in front of the mirror
Answers
\( \huge \mathrm{ Answer࿐ }\)
The correct option is :
\(\hookrightarrow\) 62 cm in back of the mirror
_____________________________
Explanation :Plane mirror always make a virtual and erect Image which is as far from the mirror as the object is !
_____________________________
\(\mathrm{ \#TeeNForeveR}\)
an elevator of mass m is initially at rest on the first floor of a building. it moves upward, and passes the second and third floors with a constant velocity, and finally stops at the fourth floor. the distance between adjacent floors is h. what is the net work done on the elevator during the entire trip, from the first floor to the fourth floor?
Answers
From the first floor to the fourth floor, the elevator's net work during the entire trip is -3mgh.
Given,
On the first floor of a building, an elevator with mass m is initially at rest. It ascends at a constant speed, passing second and third floors before coming to a stop at the fourth floor.
the separation between the floors is h
The distance from the first floor to the fourth floor Equals 3 hours of work = -mg(3 hours)
= -3mgh
Work Done = -3mgh
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What indicates that two objects are in thermal equilibrium?
Responses
The objects' temperatures are changing.
The objects' temperatures are changing.
The objects are the same size.
The objects are the same size.
The objects have the same temp
Answers
Answer:
Option (c) is the correct answer.
Explanation: When two substances does not exchange any energy with each other then they are said to be in thermal equilibrium with each other. This means the temperature of both the substances will be equal, that is why, there is no exchange of energy between them. Thus, we can conclude that when the objects have the same temperature then you can tell the two objects are in thermal equilibrium.
When a bat hits a baseball, what is transferred from the bat to the ball?
Answers
Answer:
kinetic energy
Explanation:
HELP ME PLEASE I NEED IT NOW
a) What is the value of the angle of incidence of Ray PQ on the mirror AB?
Answers
Answer:
the angle of incident is 40°
Explanation:
NQ is the normal to the mirror, therefore
angle NQA =90°
PQA = 50°
incident angle = NQA - PQA
90°- 50° = 40°
note that the angle of reflection is equal to the angle of incident
A 10 kg box is motionless on the floor. If the coefficient of static friction is 0.4 and the coefficient of kinetic friction is 0.3 (between the box and the floor), find the force required to start the block in motion
Please I need help ❤️
Answers
The force required to start the block in motion is equal to 39.2 Newton.
Given the following data:
Mass of box = 10 kgCoefficient of static friction = 0.4 Coefficient of kinetic friction = 0.3To determine the force required to start the block in motion:
Note: The force that is required to start the block in motion must be greater than or equal to the force of static friction.
Mathematically, the force of static friction is given by the formula;
\(Fs = uFn = umg\)
Where;
Fs represents the force of static friction.μ represents the coefficient of friction.\(F_n\) represents the normal force.g is the acceleration due to gravity.m is the mass of an object.Substituting the given parameters into the formula, we have;
\(F_s = 0.4 \times 10 \times 9.8\\\\F_s = 4 \times 9.8\\\\F_s = 39.2\;Newton\)
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How could a man be severely injured beig hut by some tomatoes?
Answers
Answer:
he can fall down
Explanation:
he could fall down because of the tomatoes and do notice he has fallen down because of them
A baseball with a mass of 0. 15 kilograms collides with a bat at a speed of 40 meters/second. The duration of the collision is 8. 0 x 103 seconds. The
ball moves off with a speed of 50 meters/second in the opposite direction. What is the value of the force?
Answers
The value of force is 1.7 × 10⁻³ N, with the direction opposite to that of the bat's motion.
When an object collides with another object, they exchange energy. For example, a baseball and bat collision or a car collision. When two objects collide, the force of the collision has to be equal on both sides of the collision according to Newton's Third Law. So, to find the value of force, we will apply the equation:
F = ΔP / ΔT
where F is the force, ΔP is the change in momentum, and ΔT is the time of collision. The equation represents the impulse momentum theorem.
Now, let's apply the given values to the above equation.
Final momentum (p2) = mass × final velocity (v2)
p2 = 0.15 kg × (-50 m/s)
p2 = -7.5 kg.m/s
Initial momentum (p1) = mass × initial velocity (v1)
p1 = 0.15 kg × (40 m/s)
p1 = 6 kg.m/s
Change in momentum (ΔP) = p2 - p1
ΔP = -7.5 kg.m/s - 6 kg.m/s
ΔP = -13.5 kg.m/s
Time of collision (ΔT) = 8.0 × 10³ s
Now, putting the values of ΔP and ΔT in the equation of impulse momentum theorem, we get:
F = ΔP / ΔT
F = -13.5 kg.m/s ÷ 8.0 × 10³ s
F = -1.7 × 10⁻³ N
Thus, the value of force is 1.7 × 10⁻³ N, with the direction opposite to that of the bat's motion.
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What happens to the energy in a system of magnets when the magnetic force causes two magnets to move?.
Answers
When the two magnets are made to move, the energy of the system in the system tends to change potential energy into the kinetic energy
Potential energy is the energy stored in the body while is at rest which would turn into kinetic energy when the object will initiate motion. This conversion of potential energy to kinetic energy is understood by the Work-Energy theorem. The kinetic energy is formulated as,
K.E. = 1/2 mv²
where K.E = Kinetic energy
m = mass of an object
v = velocity of the object
The work-energy theorem is stated as the products of all the forces acting on the system are equal to the difference in kinetic energies.
∑W = ΔK.E
where W = Work done on the system
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The law of reflection states that the angle of incidence and the angle of reflection will ______.Select one:a.never be equalb. always add up to 90 degreesc.always add up to 180 degreesd.always be equal
Answers
The law of reflection states that the angle of incidence and the angle of reflection will always be equal.
When a ray of light reflects off a surface, the angle of reflection is equal to the angle of incidence.