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andrew11 [14]
3 years ago
13

Let Pk denote the vector space of all polynomials with degree less than or equal to k. Define a linear transformation T : P4 ! P

3 by T(f(x)) = f(0)+f '1)(x-1)+f ''(2)(x-2)^2+f'''(3)(x-3)^3. Find the matrix representation for T relative to the standard basis {1; x; x^2; x^3; x^4} of R4 and the reversed standard basis {x^3; x^2; x; 1} of R3.
Mathematics
1 answer:
shusha [124]3 years ago
6 0

Answer:

Step-by-step explanation:

T(1)=1=0*x^3 0*x^2 0*x 1*1 T(x)=x-1=0*x^3 0*x^2 1*x (-1)*1 T(x^2)=2x^2-6x 6=0*x^3 2*x^2 (-6)*x 6 T(x^3)=6x^3-48*x^2 141*x-141 T(x^4)=24*x^3-204*x^2 628*x-604*1 collect the coefficient matrix and take its transpose

0 0 0 6 24

0 0 2 -48 -204

0 1 -6 141 628

1 -1 6 -141 -604

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Wewaii [24]

Answer:

36/25

Step-by-step explanation:

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8 0
3 years ago
A garden box has a perimeter of 18½ feet. If the length is 5 feet, what is the area of the garden box?
Travka [436]

Answer:

area=\frac{85}{4}feet^{2}

Step-by-step explanation:

1. Approach

Since it is given that the garden box is a rectangle, then the opposite sides are congruent. One can use this to their advantage, by setting up an equation that enables them to solve for the width of the rectangle. After doing so, one will multiply the width by the given length and solve for the area.

2. Solve for the width

It is given that the garden box is a rectangle. As per its definition, opposite sides in a rectangle are congruent. The problem gives the length and the perimeter of the rectangle, therefore, one can set up an equation and solve for the width.

perimeter=18\frac{1}{2}\\\\length=5

2(length+width)=perimeter

Substitute,

2(5+width)=18\frac{1}{2}

Conver the mixed number to an improper fraction. This can be done by multiplying the "number" part of the mixed number by the denominator of the fraction. Then add the result to the numerator.

2(5+width)=\frac{37}{2}

Inverse operations,

2(5+width)=\frac{37}{2}\\/2\\\\5+width=\frac{37}{4}\\-5\\\\width=\frac{17}{4}

3. Solve for the area

Now that one has solved for the width of the box, one must solve for the area. This can be done by multiplying the length by the width. Since the width is a fraction, one must remember, that when multiplying an integer by a fraction, one will multiply the integer by the numerator (the top of the fraction), and then simplify by reducing the fraction, if possible. Reducing the fraction is when one divides both the numerator and the denominator by the GCF (Greatest Common Factor).

length=5\\\\width = \frac{17}{4}

length*width=area

Substitute,

5*\frac{17}{4}\\\\=\frac{85}{4}

area=\frac{85}{4}feet^{2}

4 0
3 years ago
A company manufactures and sells shirts. The daily profit the company makes
Rom4ik [11]

Answer:

f(-5) = -150

Meaning: If the company sells -5 shirts, they would make a profit of -150 dollars. This interpretation of loss in the context of the problem

f(7) = -30

Meaning: If the company sells 7 shirts, they would make a profit of -30 dollars. This interpretation of loss in the context of the problem

f(10.5) = 5

Meaning: If the company sells 10.5 shirts, they would make a profit of 5 dollars. This interpretation of profit in the context of the problem

Step-by-step explanation:

Given

f(x) = 10x - 100

Required

Fill in the gap

f(-5) = 10 * -5 -100

f(-5) = -50 -100

f(-5) = -150

f(7) = 10 * 7 - 100

f(7) = 70 - 100

f(7) = -30

f(10.5) = 10 * 10.5 - 100

f(10.5) = 105 - 100

f(10.5) = 5

5 0
3 years ago
A bar graph shows that sports books received 9 votes . If the scale is 0 to 20 by twos, where should the bar end for the sports
Anna35 [415]

Check the picture below.

7 0
3 years ago
A triangle has a perimeter of 165 cm. The first side is 65 cm less than twice the second side. The third side is 10 cm less than
Neko [114]

Answer:A triangle has a perimeter of 165 cm. The first side is 65 cm less than twice the second side. The third side is 10 cm less than the second side. Write and solve an equation to find the length of each side of the triangle.

The sides can be found by taking the square root of the area. , where = side. . So the length of a side is 6.3 cm.

Step-by-step explanation:

3 0
3 years ago
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