Final answer:
The solutions to ax = 0 are all vectors in the null space of the matrix a. These solutions can be expressed as a parametric vector form, which may involve free variables if the matrix a has any. The general solution is a linear combination of the basis vectors for this null space.
Explanation:
The question is asking to describe all solutions to the equation ax = 0 in parametric vector form, given that a is row equivalent to a certain matrix. This matrix is understood to allow a decomposition of a vector into its perpendicular components. Such a matrix transformation typically reduces to finding the null space of the matrix.
To find the solution to ax = 0, we understand that any vector x that satisfies this equation is in the null space of a. The null vector is the most trivial solution, where all components of the vector are zero. But if there are free variables involved, the solutions may form a subspace spanned by these free variables. The general solution can then be expressed as a linear combination of these basis vectors that span the null space of a.
For instance, if our matrix a has one free variable, the parametric vector form of the solution might be t[1,0,0] if the first column of a is the pivot column and the rest are free columns. Here, t would be any scalar, representing a line through the origin in the direction of the vector [1,0,0] in three-dimensional space.
(3^-2)^4 *3^3 divided by 3^-3
Three friends rented a kayak. It cost $4 per hour per person to rent the kayak, plus $2 for each life jacket, and $3 to park the car. It cost $57 in all. How many hours did they spend kayaking? Write an equation and solve.
Write an equation that shows the equivalency between meters and gigameters.
To convert between meters and gigameters, use the equation 1 Gm = 10^9 m. This relationship is important for converting units in scientific and mathematical contexts.
Explanation:To show the equivalency between meters and gigameters, we can use the following equation:
1 gigameter (Gm) = 10⁹ meters (m).
This equation means that one gigameter is equal to one billion meters. We can use this equivalency for converting measurements from meters to gigameters or vice versa. For example, to convert 2 gigameters to meters, simply multiply 2 by 10⁹ to get 2 x 10⁹ meters.
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what is 100/6 as a decimal
Which mathematical property is demonstrated?
If w = - 8 and - 8 = u, then w = u.
Question 4 options:
symmetric property of equality
transitive property of equality
closure property of addition
closure property of multiplication
The measure of the vertex angle in an isosceles triangle is six less than four time the measure of the base angle. find the measure of the vertex angle
Rectangle ABCD is graphed in the coordinate plane. The following are the vertices of the rectangle: A(5, 1), B(7,1) C(7, 6), and D(5, 6). What is the perimeter of rectangle ABCD?
The perimeter of the rectangle ABCD is 14 units after using the distance formula.
What is a rectangle?It is defined as the two-dimensional geometry in which the angle between the adjacent sides is 90 degrees. It is a type of quadrilateral.
The area of a rectangle can be calculated using the following formula:
Rectangle area = length x width
The missing figure is in the picture, please attached to the picture.
We have:
Rectangle ABCD is graphed in the coordinate plane.
As we know,
The distance formula: it can be defined as the formula for finding the distance between two points. It has given the shortest path distance between two points.
The distance formula can be given as:
[tex]\rm d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
After using the distance formula:
The perimeter of the rectangle = AB + BC + CD + DA
The perimeter of the rectangle = 2 + 5 + 2 + 5
The perimeter of the rectangle = 14 units
Thus, the perimeter of the rectangle ABCD is 14 units after using the distance formula.
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4(-3)(15/3)
20 points
You are making identical gift bags using 24 candles and 36 bottles of lotion .what is the greatest number of gift bags you can make with no items left over. Show your work
Answer: 12
Step-by-step explanation:
Given : You are making identical gift bags using 24 candles and 36 bottles of lotion .
Then, the greatest number of gift bags you can make with no items left over= greatest common factor of 24 and 36
Prime factorization of 24 and 36 :_
[tex]24=2\times2\times2\times3\\\\ 36=2\times2\times3\times3[/tex]
GCF(24 , 36)= [tex]2\times2\times3=12[/tex]
Hence, the greatest number of gift bags you can make with no items left over =12
Enter an inequality that represents the graph in the box.
PLEASE HELP ME!
its actually y=>3x-1 because it's shaded below the line.
below the line: < or <=
above the line: > or>=
Victor is walking at a rate of 1 mile every 17 minutes. sarah is walking at a rate of 1 mile every 14 minutes. if they are 10 miles apart and are approaching each other along a straight road, how many hours will it take them to meet, rounded to the nearest hundredth?
64 ounces in 8 cups unit rate
What is the measure of angle S?
132 + 56 + 79 = 267
360 - 267 = 93
answer is 93 degrees
For what values of x is the quantity x−9 negative?
The value of x is the quantity x−9 negative would be negative for all values of x less than 9.
What is algebra?Algebra is the analysis of mathematical representations, while logic is the manipulation of those symbols.
What is inequality?
Inequality is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are not equal.
To determine the values of x is the quantity x−9 negative
As per the given condition, translate the phrase into an algebraic expression:
⇒ x - 9 < 0
Add 9 to both sides of the inequality,
⇒ x - 9 + 9 < 9
⇒ x < 9
So x - 9 is negative for all values of x less than 9.
Hence, the value of x is the quantity x−9 negative would be negative for all values of x less than 9.
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What is the slope-intercept form of a linear equation? explain why this form is called the slope-intercept form?
Answer: The slope intercept form of a linear equation has the following form where the equation is solved for y in terms of x: y = a + bx. b is the slope. a is a constant term. It is the y intercept, the place where the line crosses the y axis.
Hope this helps:)
1) if the sum of two prime numbers is divisible by 7 and is a perfect square, what is a possible product of these numbers?
2) If it takes 5 pipes 2 hours to fill 2 pools and they output water at the same rate, how many hours will it take 1 pipeto fill 1 pool?
3) On a test with 30 questions, Ron got 6 questions wrong. What is his score expressed as a percentage?
4)35 percent of a class voted to go to the museum. If 7 students chose to go to the museum, how many students are there in the class?
5)Katie is 2 years younger than her brother. In 12 years, her brother will be twice as old as she is now. How old is Katie?
1) There are no prime numbers whose sum is divisible by 7 and is a perfect square.
2) 1 pipe will take 2 hours to fill 1 pool, as it fills at the same rate as all 5 pipes combined.
3) Ron's score is 20%.
4) There are 20 students in the class.
5) Katie is 14 years old.
Let's evaluate each of the statements you provided:
1) The sum of 2 prime numbers is ALWAYS even, so if x and y are prime numbers, then (x+y)/7 = z².
In this case either x = 2 or y = 5 (or vice versa). This is the only possibility to be divisible by 7.
Then (5+2)/7 = 1² = 1.
The first part is correct; the sum of two prime numbers is always even. However, the statement about (x+y)/7 = z² doesn't hold true for all prime numbers. It might work for the specific values of x and y you mentioned, but it's not a general rule.
5 pipes → 2 hours → 2 pools, then 1 pipe takes (5x2) 10 hours to fill 2 pools, or 1 pipe takes (10/2) 5 hours to fill 1 pool.
This is a correct interpretation of the information provided. If 5 pipes can fill 2 pools in 2 hours, then each pipe alone takes 10 hours to fill 2 pools, or 5 hours to fill 1 pool.
It's 6/30 = 0.2 or 20%.
Correct, 6/30 simplifies to 0.2, which is equivalent to 20%.
35% represents 7 students, and 100% of the class represents x students. Therefore, 35%/7 = 100%/x. Cross-multiplying, we get 0.35x = 7, and x = 7/0.35, which simplifies to x = 20 students.
Let K be the age of Katie, and B her brother's age. K + 2 = B (Today). In 12 years, B+12 will be twice K (remember K = B-2):
B+12 = 2(K) OR B+12 = 2(B-2)
B+12 = 2(B-2)
B+12 = 2B - 4
12 + 4 = B, and B = 16. If the brother is 16, then Kate is 14.
The calculations and logic are correct. If Katie's brother is 16, then Katie is indeed 14 years old.
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If ab= 37 and xy =1/37, what is the value of the product x x b x y x a
In this case, we are simply asked to find for the value of:
xbya or abxy
Since we know that given values of ab and xy, therefore abxy is:
abxy = 37 * (1 / 37)
abxy = 37 / 37
abxy = 1
What is 7 1/5 written as a percent
The circumference of a circle is 25.12 yards. What is the radius of the circle?
Ali currently has $25. He is going to start saving $5 every week.
Which equation represents this situation?
y = 25x – 5
y = 5x + 25
y = 25x + 5
y = 5x – 25
Solve for x. 45(15x+20)−7x=56(12x−24)+6
Equations may be true or false, just as word sentences may be true or false. The equation:
3 + x = 7
will be false if any number except 4 is substituted for the variable. The value of the variable for which the equation is true (4 in this example) is called the solution of the equation. We can determine whether or not a given number is a solution of a given equation by substituting the number in place of the variable and determining the truth or falsity of the result.
Example 1 Determine if the value 3 is a solution of the equation
4x - 2 = 3x + 1
Solution We substitute the value 3 for x in the equation and see if the left-hand member equals the right-hand member.
4(3) - 2 = 3(3) + 1
12 - 2 = 9 + 1
10 = 10
Ans. 3 is a solution.
The first-degree equations that we consider in this chapter have at most one solution. The solutions to many such equations can be determined by inspection.
Example 2 Find the solution of each equation by inspection.
a. x + 5 = 12
b. 4 · x = -20
Solutions a. 7 is the solution since 7 + 5 = 12.
b. -5 is the solution since 4(-5) = -20.
Problem page the perimeter of a rectangular garden is 274 feet. if the width of the garden is 63 feet, what is its length?
20 point question!!! Please help me! Number 4.
If m is the midpoint of pq, what is the relationship between pm and pq
Midpoint, as the word suggests, means the point which lies in the middle of something. If m is the midpoint of PQ then PM will be half of PQ.
What does a midpoint mean?Midpoint, as the word suggests, means the point which lies in the middle of something. The midpoint of a line segment means a point that lies in the mid of the given line segment.
Given that m is the midpoint of PQ. Therefore, it will divide PQ into two equal parts. Therefore, we can write,
PQ = PM + MQ
But since PM = MQ,
PQ = PM + PM
PQ = 2 PM
PM = PQ/2
Hence, if m is the midpoint of PQ then the relationship between PM and PQ will be half of PQ.
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How can you find the measure of a distance that you cannot measure directly?
Help! I'm not sure how you would graph x = -6. There's no other information available for this question.
ABCD is a rectangle. The length of BD is 8 units, and m angle ABD is 67°. The length of AC is units, and m angle CDB is °.
Answer:
cbd is 23 degrees
Step-by-step explanation:
The total amount of money in cents in 3x nickels 4x dimes and 3x quarters
1.30 or 130
Step-by-step explanation:
Twice the number n is no less than 10 units from -1
The mathematical equation 'Twice the number n is no less than 10 units from -1' can be solved as two inequalities, yielding a solution of n ≤ -5.5 or n >= 4.5.
Explanation:This is a math problem asking about the absolute value inequality of an equation. The question states 'Twice the number n is no less than 10 units from -1'. This can be written as |2n + 1| >= 10. To solve it, we'll divide it into two inequalities: 2n + 1 >= 10 and -2n - 1 >= 10. Solving the first inequality yields n >= 4.5, and solving the second inequality yields n <= -5.5. Thus, the solution to the equation is n ≤ -5.5 or n >= 4.5.
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Write the point-slope form of the equation of the line with a slope of - 2 and an x-intercept of - 1. Include your work in your final answer.
Please helps and show how.
Answer:
y= -2x-2
Step-by-step explanation:
point slope- y=mx+b
y-y1=m(x-x1)
you have the point (-1,0) and you have to use (x1, y1)
your slope is -2
Substitute
y-0= -2(x +1)
Distribute
y-0=-2x-2
+0 +0
y= -2x-2