The nullity of the standard matrix for T is the dimension of its null space, which is 1.
To write the standard matrix for T, we need to find the image of the standard basis vectors for R2. Thus, we have:
T(1,0) = (1+3(0),1-0,1) = (1,1,1)
T(0,1) = (0+3(1),0-1,0) = (3,-1,0)
Therefore, the standard matrix for T is:
| 1 3 |
|-1 -1|
| 1 0 |
To find the column space of this matrix, we need to find all linear combinations of its columns. The first column can be written as (1,-1,1) plus 2 times the third column, so the column space is spanned by (1,-1,1) and (0,-1,0).
The rank of the standard matrix for T is the dimension of its column space, which is 2.
To find the null space of the standard matrix for T, we need to solve the equation Ax=0, where A is the standard matrix. This gives us the system of equations:
x1 + 3x2 = 0
-x1 - x2 = 0
The general solution to this system is x1=-3x2, x2=x2, so the null space is spanned by (-3,1).
This means that there is only one linearly independent solution to Ax=0, and that the dimension of the domain of T minus the rank of the standard matrix equals the nullity.
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There is one free variable (x2) in the null space, the nullity of A is 1.
1. To find the standard matrix for the linear transformation T(x1, x2) = (x1 + 3x2, x1 - x2, x1), arrange the coefficients of x1 and x2 in columns:
A = | 1 3 |
| 1 -1 |
| 1 0 |
2. The column space of the standard matrix A is the span of its columns:
Col(A) = span{ (1, 1, 1), (3, -1, 0) }
3. The rank of a matrix is the dimension of its column space. To find the rank of A, row reduce it to echelon form:
A ~ | 1 3 |
| 0 -4 |
| 0 0 |
Since there are two nonzero rows, the rank of A is 2.
4. To find the null space of A, we solve the homogeneous system Ax = 0:
| 1 3 | | x1 | = | 0 |
| 1 -1 | | x2 | = | 0 |
| 1 0 | | 0 |
From the system, x1 = -3x2. The null space of A is:
N(A) = { (-3x2, x2) : x2 ∈ R }
5. The nullity of a matrix is the dimension of its null space. Since there is one free variable (x2) in the null space, the nullity of A is 1.
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What is the relation between ∆ H and ∆ U under what condition the two ate are equal?
Therefore , the solution of the given problem of expressions comes out to be ΔH = ΔU.
By changing each variable with the a number and running arithmetic operations, an algebraic equation must be checked. Since 79 + 8 = 87 , the answer variable in the aforementioned example is the same as 6. We can substitute the variables values with those we know in order to evaluate the expression.
Here,
Given:
During a procedure that takes place in a closed vessel, ΔH = ΔU.
ΔV = 0 denotes that there are no gaseous reactants or products in the reaction, or
that the amount of moles for gaseous products equals the number of molecules of gaseous reactants.
Therefore , the solution of the given problem of expression comes out to be ΔH = ΔU.
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Is triangle FHK similar to triangle GHJ? If so, which postulate or theorem proves these two triangles are similar?
The postulate or theorem that proves that the two triangles are similar is: ΔFHK is similar to ΔGHJ by the SSS Similarity Theorem.
Properties of similar triangles.When the corresponding sides or measure of angles of two or more triangles have common properties, the triangles are said to be similar. This can be deduced on comparing the corresponding length of sides or measure of angles.
In the given question, comparing the corresponding sides of the triangle. We have;
HG/ HF = 10/ 22
= 0.4545
Also,
HJ/ HK = 15/ 33
= 0.4545
Since the two ratios are the same, it implies that the third sides of the two triangles would give the same answer. Thus they are similar.
Therefore, the postulate or theorem that proves that the two triangles are similar is: ΔFHK is similar to ΔGHJ by the SSS Similarity Theorem.
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Nick and Chloe left their campsite by canoe and paddled
downstream at an average speed of 12 km/h. They
paddled downstream for 2 km. How much time did the
campers take to get downstream? (include formula)
To find out how much time the campers took to get downstream, we can use the formula:
time = distance / speed
where distance is the distance they paddled downstream (in this case, 2 km) and speed is their average speed (in this case, 12 km/h). So we can substitute in the values we know to get:
time = 2 km / 12 km/h
To solve for the time, we can divide the distance by the speed:
time = 2/12 hours = 0.17 hours or approximately 10.2 minutes
So it takes 10.2 minutes for Nick and Chloe to get downstream.
How do you find the number of functions from class 11 to B?
Number of functions. It is an important topic in sets and relation. A relation in which each input has a specific output is called a function. If f is a function from set A to set B, then each element of A will be plot with only one element in B.
1. Number of possible functions: If a set A has j elements and set B has k elements, then the number of functions possible from A to B is j ^k.
2. Number of Injective Functions (One to One): If set A has j elements and set B has k elements, j ≥k, then the number of injective functions or one to one function is given by j! / (j-k)!
3. Number of Bijective functions: If there is bijection between two sets A and B, then both sets will have the same number of elements. If n(A) = n(B) = m, then number of bijective functions = m!
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Billy bought a lamp that costs $13.00 the tax is 5% what is the amount of tax he will have to pay
Answer:
0.65
Step-by-step explanation:
[tex]13.00*5%=0.65[/tex]
Answer:
Price before tax: $ 13.00
+ Sales tax (5%): $ 0.65
Total price including tax: $ 13.65
Good luck <33
Noah ia taking a multiple choice tet with a total of 20 point available. Each quetion i worth exactly 1 point. What would be Noah' tet core (out of 20) if he got 3 quetion wrong? What would be hi core if he got quetion wrong?
Noah's total score is 17 out of 20 since he answered 3 questions wrong.
A word problem is a short passage presenting a "real-life" situation where an issue has to be solved using math.
Word problems are seen as an essential component of learning in the elementary curriculum because they require students to apply their understanding of various concepts to situations that are representative of "real-life" situations.
Given that,
Total multiple choice questions = 20
The score of answering one question correctly = 1
Total number of questions Noah wrongly answered = 3
Therefore, his score out of 20 = 20 - 3 = 17
Thus, Noah's total score is 17 out of 20 since he answered 3 questions wrong.
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Find the area of the polygon with the given vertice. N(-2,1), P(3,1), Q(3,-1), R(-2,-1)
The area of polygon is 10 units.
To find the area of the polygon with the given vertices, we can use the shoelace formula, which is a method for finding the area of a polygon when given its vertices in a certain order.
The shoelace formula is:
Area = (1/2) * |(x1y2 + x2y3 + x3y4 + ... + xn-1yn + xny1) - (y1x2 + y2x3 + y3x4 + ... + yn-1xn + ynx1)|
where (x1, y1), (x2, y2), (x3, y3), ..., (xn, yn) are the vertices of the polygon in a certain order.
For the given vertices, we have:
N(-2,1), P(3,1), Q(3,-1), R(-2,-1)
So the area of the polygon is:
Area = (1/2) * |(-21 + 3-1 + 3*-1 + -21) - (13 + 13 + -1-2 + -1*-2)|
Area = (1/2) * |(-2 - 3 - 3 + 2) - (3 + 3 + 2 + 2)|
Area = (1/2) * |-8 - 12|
Area = (1/2) * 20
Area = 10 square units
So the area of the polygon is 10 square units.
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I DIDNT HAVE MUCH POINTS IM SORRY! CAN ANYONE PLEASE ANSWER :(
Which statement about determining the quotient 16÷12 is true?
Responses
Because 4×12=16, 16 divided by 12 is 4.
Because , 4 times 1 half equals 16, , 16 divided by , 1 half, is 4.
Because 24×12=16, 16 divided by 12 is 24.
Because , 32 times 1 half equals 16, , 16 divided by , 1 half, is 24.
Because 8×12=16, 16 divided by 12 is 8.
Because , 8 times 1 half equals 16, , 16 divided by , 1 half, is 8.
Because 32×12=16, 16 divided by 12 is 32
Answer:1.33333333 so B
Step-by-step explanation:16/12 = 1.33333333
xena runs halfway across a field in 1 minute. the next fourth of the field takes her 2/3 of a minute to cross, the next eighth takes 4/9 of a minute, and so on, with each half of the previous distance taking 2/3 of the previous time. how many minutes does it take xena to cross the field?
it would take 3 minutes for Xena to finish the run.
What is the geometric sequence?A geometric sequence is a sequence of numbers that follows a pattern where the next term is found by multiplying by a constant called the common ratio, r.
From the general formula of the nth term of the geometric sequence:
Aₙ = A rⁿ⁻¹
Given, every stop Xena makes is diminished by half of the distance she covers from the preceding. And she takes 2/3 of the time she used from the previous stop.
The problem above is a WORDED PROBLEM that involves INFINITE GEOMETRIC PROGRESSION.
As we all know, a worded problem just basically needs a thorough understanding of what is happening on the problem to translate words and phrases into algebraic phrases and equations.
Meanwhile, a geometric progression is a type of progression in which the proceeding value is greater or smaller than the previous by a ratio. Typical geometric progression problems involve ratios of greater than 1.
The above problem is an “infinite geometric progression” in which the ratio is less than 1.
Refer to the illustration below.
We first take elimination of the given.
Since both the distance and the time have infinite number of stops to complete the run, the SUM OF INFINITE GEOMETRIC PROGRESSION is
S = (A₁)/(1-r)
a1 = first term
r = ratio
Substituting the values, we have
S = (1min) / (1- (2/3))
S = 1 / (1/3)
S = 3 minutes
Therefore, it would take 3 minutes for Xena to finish the run.
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the ___ is the line through the vertices of an ellipse.
Answer: The major axis is the line segment that passes through the foci of the ellipse. The endpoints of this axis are called the vertices of the ellipse. The segment passing through the center and perpendicular to the major axis is called the minor axis.
Question 7
SPORTS The Fan Cost Index (FCI) tracks the average costs for attending sporting events, including tickets, drinks, food, parking, programs,
and souvenirs. According to the FCI, a family of four would spend a total of $592.30 to attend two Major League Baseball (MLB) games and
one National Basketball Association (NBA) game. The family would spend $691.31 to attend one MLB and two NBA games.
a. Write a system of equations to find the family's costs for each kind of game according to the FCI. Let x be the cost of MLB games and y be
the cost of NBA games.
MLB: $
x+
NBA: $
y=592.30
b. Solve the system of equations to find the cost for a family of four to attend each kind of game according to the FCI.
y=
Answer: a. To write a system of equations to find the family's costs for each kind of game according to the FCI, we can use the information provided in the problem. According to the FCI, a family of four would spend a total of $592.30 to attend two MLB games and one NBA game, and the family would spend $691.31 to attend one MLB and two NBA games. Let x be the cost of MLB games and y be the cost of NBA games. Then we can write the following system of equations:
x + y = 592.30 equation 1 represents the cost of attending two MLB games and one NBA game
2y + x = 691.31 equation 2 represents the cost of attending one MLB and two NBA games
b. We can solve the system of equations to find the cost for a family of four to attend each kind of game according to the FCI.
We can substitute equation 1 into equation 2
2y + (592.30 - y) = 691.31
2y + 592.30 - y = 691.31
2y = 691.31 - 592.30 = 99.01
y = 49.51
we know y represents cost of NBA games so 49.51 is cost of one NBA game for a family of four
x = 592.30 - y = 592.30 - 49.51 = 542.79
x represents cost of MLB games so 542.79 is the cost of one MLB game for a family of four
So, according to the FCI, a family of four would spend $542.79 to attend one MLB game, and $49.51 to attend one NBA game.
Step-by-step explanation:
Assume an exponential function has a starting value of 16 and a decay rate of 44%. Write an equation to model the situation.
16 x 0.56^x
where 16 is the sfadtin value
0.56 is how much the amount is decreasing by each time
x= number of hours/days etc
For what value of c for which pair of linear equations cx y 2 and 6x 2y 4 will have infinitely many solutions?
For the linear equations, cx – y = 2 and 6x – 2y = 4 will have infinitely many solutions; the value of c will be 3.
cx – y = 2 eqn(1)
6x – 2y = 4 eqn(2)
The condition for infinitely many solutions is
a₁/a₂ = b₁/b₂ = c₁/c₂ eqn(3)
Here, a₁ = c, b₁ = −1, c₁ = −2 and a₂ = 6, b₂ = −2, c₂ = −4
From eqn(3), we get
c/6 = −1/−2 = −2/−4
⇒c/6 = 1/2 and c/6 = 2/4
⇒c = 3 and c = 3
Hence, for the pair of equations that will have infinitely many solutions, the value of c will be 3.
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Is a triangle possible with length of the sides as 10.2 cm 5.8 cm and 4.5 cm?
The Triangle with length of the sides as 10.2 cm 5.8 cm and 4.5 cm is possible.
What is Triangle Inequality Theorem?Triangle inequality is a theorem in Euclidean geometry that states any two triangle sides added together must be greater than or equal to the third side; it is represented by the symbols a + b ≥ c. The basic tenet of the theorem is that a straight line connects any two places.
Given:
Sides of triangle is: 10.2 cm 5.8 cm and 4.5 cm
Now, using Triangle inequality "the sum of two sides is greater than the third side of the triangle."
Suppose such a triangle is possible.
4.5 + 5.8 > 10.2
5.8 + 10.2 > 4.5
10.2 + 4.5 > 5.8
Hence, the triangle is possible.
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What is 2 with an exponent of 0?
Therefore, 2 with an exponent of 0 is equal to 1.
When a number is raised to an exponent of 0, the result is 1. This applies to any non-zero number, including 2. Therefore, 2 with an exponent of 0 is equal to 1.
The reason is that any number raised to the power of 0 is 1. This is because any number multiplied by 1 is itself, and any number raised to the power of 1 is itself. Therefore, raising a number to the power of 0 is the same as multiplying it by 1, so the result is 1.
It is also important to note that a number raised to the power of 1 is the number itself, so 2^1=2, 2^2=4, 2^3=8 and so on.
Therefore, 2 with an exponent of 0 is equal to 1.
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When asked to rewrite $364 + 728 as a product of the GCF and the sum of two numbers a student wrote 364 (1 + 2) is the student correct why or why not?
Answer: The student got the wrong answer. ✅
Step-by-step explanation: When you add 364 and 728 together, you get 1092. To make it easier to understand, it is like having 3 piles of 100, 6 piles of 10, and 4 ones on one side, and 7 piles of 100, 2 piles of 10, and 8 ones on the other side, when you put them together you get 10 piles of 100, 8 piles of 10 and 12 ones.
To find the biggest number they have in common, you can divide both numbers by 2 and you get 182 and 364.
So the correct way to rewrite 364 + 728 is 364 * (3+2) ✅
Is 2 3 4 5 6 6 a function or not?
No, 2 3 4 5 6 6 is not a function.
A function is a set of ordered pairs (x, f(x)) where each input value x corresponds to exactly one output value f(x). In other words, for any given input value x, there should be only one corresponding output value f(x).
2 3 4 5 6 6 is a list of integers, but it does not represent any ordered pairs. It is not clear which integers correspond to input values and which correspond to output values. Therefore, it cannot be considered a function.
It's important to note that in order to identify if a set of ordered pairs is a function or not, you must check if there's any value that is repeated as an input value, if so then it's not a function.
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What is the center theorem?
The central law expresses that even if a group is not normally distributed, the means of sufficiently large samples drawn from that group will be normally distributed.
A key conclusion of graph theory is known as the Center Theorem, which asserts that in a connected graph, the minimal degree of a vertex's degree is equal to the most significant number of vertices that can be removed from the graph while maintaining connectivity.
Generally, it is the collection of vertices with the lowest eccentricity that constitutes the centre of a connected graph, where the eccentricity of a vertex is defined as the most significant distance between that vertex and any other vertex in the graph. The radius of a graph is the number of vertices that make up its centre.
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Look at the following graph of the given equation. Points marked on graph: (1,0), (0,2), (1,4), (2,6). Determine whether the equation is a function. Explain why or why not. y = 2x + 2
The equation y = 2x + 2 is a function. This can be determined by observing that for each x-value, there is exactly one corresponding y-value. In other words, the graph of the equation passes the vertical line test, which states that a graph represents a function if no vertical line intersects the graph more than once.
Additionally, as the given points (1,0), (0,2), (1,4), (2,6) all fall on the same line, all with a slope of 2 and y-intercept of 2, so this confirms that the equation is indeed a function.
Help fast please help
As the difference between the output values, the rate of change of the linear equation is 4/3.
what is linear equation ?A linear equation is one that has the form y=mx+b in algebra. B is the slope, and m is the y-intercept. It's usual to refer to the previous clause as a "linear equation with two variables" because y and x are variables. The two-variable linear equations known as bivariate linear equations. There are several instances of linear equations: 2x - 3 = 0, 2y = 8, m + 1 = 0, x/2 = 3, x + y = 2, and 3x - y + z = 3. It is referred to as being linear when an equation has the form y=mx+b, where m stands for the slope and b for the y-intercept.
given
The specified points' coordinates should be found.
The difference between the output values = 4
The difference between the input values = 3
Find the expression for the ratio of the change in y to the change in x: yx.
= 4/3
As the difference between the output values, the rate of change of the linear equation is 4/3.
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Lori works as a cartoonist for a teen magazine. The time she spends sketching is given by the equation m = 12s, where m is the number of minutes and s is the number of sketches.
If Lori made
3
4
of a sketch
If Lori made a fraction of 3/4 of a sketch, then she spent 9 minutes of sketching.
What is fraction?A fraction is a portion of something. Mathematics refers to a number as a quotient when the numerator and denominator are divided. Both are integers in a simple fraction. There is a fraction in the numerator or denominator of a complex fraction. The numerator and denominator of a proper fraction are different sizes.
Given the equation
m = 12s,
where
m is the number of minutes, and
s is the number of sketches.
Lori made 3/4 of a sketch i.e.
m = 12(3/4)
m = 9 mins
Thus, If Lori made 3/4 of a sketch, she spent 9 minutes sketching.
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A factory produces 350 smartphones per hour. How many smartphones can it produce in 4 days, given 24 hours in a day?
A day has 24 hours, so 4 days have 4 * 24 = 96 hours.
The factory produces 350 smartphones per hour, so in 96 hours it can produce 350 * 96 = 33600 smartphones.
Therefore the factory can produce 33600 smartphones in 4 days.
what is angle LOM equal to?
Answer:
90 degrees
Step-by-step explanation:
-2y - 5 > 5/2 problem 4 Record Practice Journal (red)
On solving the provided question, we can say that the inequality equation is -2y - 5 > 5/2 => -4y - 10 > 5 => -4y > 15 +. y > -15/4
What is inequality?An inequality in mathematics is a relationship between two expressions and or values that is not equal. Thus, imbalance leads to inequality. An inequality creates the link between two values that are not equal in mathematics. Egality is distinct from inequality. When two values are not equal, most commonly use the not equal sign (). Different inequalities are used to contrast values, no matter how little or large. Many simple inequalities may be resolved by modifying the two sides until the variables are all that remain. But a number of things contribute to inequality: Negative values on both sides are divided or added. Trade off the left and right.
here,
the inequality equation is
-2y - 5 > 5/2
-4y - 10 > 5
-4y > 15
y > -15/4
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Find the value of x in the figure. Two intersecting lines with vertical angles x degrees and 40 degrees. (1 point)
The answer to thus question is 140 degrees.
To solve this problem we must know about vertical angles,
Vertical angles are always congruent (congruent means having the same measure or size). Therefore, if one angle measures x degrees, the other angle also measures x degrees. If we know that one of the angles measures 40 degrees, we can use that information to find the measure of the other angle, which is x.
Since the angles are vertical angles, the sum of both angles will be equal to 180 degrees, therefore
x + 40 = 180
x = 140
Therefore, the value of x is 140 degrees.
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Study the equation.
2(x+6)=mx+12
Which value of m results in the equation having infinitely many solutions?
Responses
m=0m is equal to 0
m=2m is equal to 2
m=6m is equal to 6
m=12
The value of m = 2 results in the given equation having infinitely many solutions.
What is an Equation?An equation is the mathematical statement of two expressions situated on two sides connected with an equal to sign.
The two sides of an equation is usually called as left hand side and right hand side.
Given the equation,
2(x + 6) = mx + 12
Applying the distributive property a(b + c) = a b + a c on the left hand side,
2x + 12 = mx + 12
Two equations have infinitely many solutions when both the equations are same.
Left and right side are equal if m = 2.
Hence there are infinitely many solutions when m = 2.
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A number is increased by 7, twice the result is 41.what is the number?
Answer:
After increasing it by 7, it becomes = 2x + 7. => x = 82/2 = 41. Hence, the original number = 41.
What is the y-intercept of the line given by the equation below y =- 10x 14?
The y-intercept of the line given by the equation y = -10x + 14 is (0, 14)
We know that the y-intercept of the line is nothing but the point at which the line intersects the y-axis.
To find the y-intercept, we substitute x = 0 in given equation of line and solve it for y.
In this question we have been given an equation of line y = -10x + 14
to find y-intercept substitute x = 0 in given equation.
y = -10x + 14
y = -10(0) + 14
y = 0 + 14
y = 14
This means, a line y = -10x + 14 intersects the y-axis at point (0, 14)
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I have 2 questions
1. 7/8n³ + 2 -1/2n³
2. 1.4y² + 4.6 +8y - 0.5 -4y + y2
The simplified expressions are given as follows:
1. 3n³/8 + 2.
2. 2.4y² + 7.6y - 0.5.
How to simplify the expression?The expressions are simplified combining the like terms.
The first expression is given as follows:
(7/8)n³ + 2 - (1/2)n³.
The like terms are given as follows:
(7/8)n³ - (1/2)n³ = 7n³/8 - 4n³/8 = 3n³/8.2.Hence the simplified expression is:
3n³/8 + 2.
The second expression is defined as follows:
1.4y² + 3.6y + 8y - 0.5 - 4y + y².
The like terms are given as follows:
1.4y² + y² = 2.4y².3.6y + 8y - 4y = 7.6y.-0.5.Hence the simplified expression is:
2.4y² + 7.6y - 0.5.
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Do the sides 7cm 24cm and 25cm form a right-angled triangle give reasons?
The sides 7 cm, 24 cm and 25 cm can form a right-angled triangle. The Pythagorean theorem can be used to verify this.
From Pythagorean theorem, it can be said that, the square of hypotenuse side is equal to the sum of other two sides of a right angles triangle.
Let AB, CA and BC be the sides of a right triangle. CA is the hypotenuse side, AB and BC are the base and height sides of the right triangle. The equation can then be expressed as,
CA² = AB² + BC²
Now, let us check this rule for the above given sides of the triangle.
Longest side² = Sum of squares of other two sides
25² = 24² + 7²
625 = 625
Thus, the sides 7 cm, 24 cm and 25 cm can form a right-angled triangle.
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