Question

In: Advanced Math

Question 1 A) Show that the functions y1(t) = 1 + t 2 ; y2(t) =...

Question 1

A) Show that the functions y1(t) = 1 + t 2 ; y2(t) = 1 − t 2 are linearly independent directly from the definition of linear independence.

B)Find three functions y1(t), y2(t), y3(t) such that any two of them are linearly independent but three of them are not linearly independent.

Solutions

Expert Solution

a) Suppose that are such that . Then

implies

Second equation gives , and first equation gives

Thus, we have proved that if and only if . Thus, are linearly independent.

b) Let us consider as above, and . We know that are linearly independent by part a).

Proof of the linear independence of :

Suppose that are such that . Then

implies

Solving, we get

Thus, we have proved that if and only if . Thus, are linearly independent.

Proof of the linear independence of :

Suppose that are such that . Then

implies

Solving, we get

Thus, we have proved that if and only if . Thus, are linearly independent.

However, are not linearly independent because

Colby Messinger answered 3 years ago
Next > < Previous

Related Solutions

Two waves y1(x,t) and y2(x,t) propagate in the air: y1 = A sin(6x - 12t) y2...
Two waves y1(x,t) and y2(x,t) propagate in the air: y1 = A sin(6x - 12t) y2 = A sin(5x - 10t) Find: a) The equation of the resulting pulse, y1 + y2. b) The distance between 2 consecutive zeros in the elongation. c) The distance between 2 consecutive absolute maxima of the elongation.
Let Y1 and Y2 have joint pdf f(y1, y2) = (6(1−y2), if 0≤y1≤y2≤1 0, otherwise. a)...
Let Y1 and Y2 have joint pdf f(y1, y2) = (6(1−y2), if 0≤y1≤y2≤1 0, otherwise. a) Are Y1 and Y2 independent? Why? b) Find Cov(Y1, Y2). c) Find V(Y1−Y2). d) Find Var(Y1|Y2=y2).
a) The functions y1 = x^2 and y2 = x^5 are two solutions of the equation...
a) The functions y1 = x^2 and y2 = x^5 are two solutions of the equation x^2 y ″ − 6 x y ′ + 10 y = 0. Let  y be the solution of the equation x^2 y ″ − 6 x y ′ + 10 y = 3 x^5 satisfyng the conditions y ( 1 ) = 0 and  y ′ ( 1 ) = 1. Find the value of the function  f ( x ) = y ( x )...
The parametric equations x = x1 + (x2 − x1)t,    y = y1 + (y2 − y1)t...
The parametric equations x = x1 + (x2 − x1)t,    y = y1 + (y2 − y1)t where 0 ≤ t ≤ 1 describe the line segment that joins the points P1(x1, y1) and P2(x2, y2). Use a graphing device to draw the triangle with vertices A(1, 1), B(4, 3), C(1, 6). Find the parametrization, including endpoints, and sketch to check. (Enter your answers as a comma-separated list of equations. Let x and y be in terms of t.)
Show that the set ℝ2R2, equipped with operations (?1,?1)+˜(?2,?2)=(?1+?2+1,?1+?2−1)(x1,y1)+~(x2,y2)=(x1+x2+1,y1+y2−1) ? ⋅˜ (?,?)=(??+?−1,??−?+1) (1)defines a vector space...
Show that the set ℝ2R2, equipped with operations (?1,?1)+˜(?2,?2)=(?1+?2+1,?1+?2−1)(x1,y1)+~(x2,y2)=(x1+x2+1,y1+y2−1) ? ⋅˜ (?,?)=(??+?−1,??−?+1) (1)defines a vector space over ℝR. (2)Show that the vector space ?V defined in question 1 is isomorphic to ℝ2R2 equipped with its usual vector space operations. This means you need to define an invertible linear map ?:?→ℝ2T:V→R2.
We define a relation ∼ on R^2 by (x1,y1)∼(x2,y2) if and only if (y2−y1) ∈ 2Z....
We define a relation ∼ on R^2 by (x1,y1)∼(x2,y2) if and only if (y2−y1) ∈ 2Z. Show that the relation∼is an equivalence relation and describe the equivalence class of the point (0,1).
2. This question relates to the Fisher Intertemporal Consumption Model. Suppose Y1 = 200, Y2 =...
2. This question relates to the Fisher Intertemporal Consumption Model. Suppose Y1 = 200, Y2 = 250 and r = 0.03. a) Calculate the maximum amount of consumption in periods 1 and 2. Explain why these two values are different. b) Suppose r increases to 0.05, calculate the maximum amount of consumption in periods 1 and 2. Explain intuitively why the consumption values changed. c) Suppose Y1 increases to 300. Use r = 0.03 to answer this question. Explain numerically...
Let c(y1, y2) = y1 + y2 + (y1y2)^ −(1/3). Does this cost function have economies...
Let c(y1, y2) = y1 + y2 + (y1y2)^ −(1/3). Does this cost function have economies of scale for y1? What about economies of scope for any strictly positive y1 and y2. Hint, economies of scope exist if for a positive set of y1 and y2, c(y1, y2) < c(y1, 0) + c(0, y2). [Hint: Be very careful to handle the case of y2 = 0 separately.]
Determine which of the following pairs of functions y1 and y2 form a fundamental set of...
Determine which of the following pairs of functions y1 and y2 form a fundamental set of solutions to the differential equation: x^2*y'' - 4xy' + 6y = 0 on the interval (0, ∞). Mark all correct solutions: a) y1 = x and y2 = x^2 b) y1 = x^2 and y2 = 4x^2 c) y1 = x^2 and y2 = x^3 d) y1 = 4x^2 and y2 = x^3 e) y1 = [(x^2)+(x^3)] and y2= x^3
The joint density of Y1, Y2 is given by f(y) = k, −1 ≤ y1 ≤...
The joint density of Y1, Y2 is given by f(y) = k, −1 ≤ y1 ≤ 1, 0 ≤ y2 ≤ 1, y1 + y2 ≤ 1, y1 − y2 ≥ −1, 0, otherwise a. Find the value of k that makes this a probability density function. b. Find the probabilities P(Y2 ≤ 1/2) and P(Y1 ≥ −1/2, Y2 ≤ 1/2 c. Find the marginal distributions of Y1 and of Y2. d. Determine if Y1 and Y2 are independent e....
ADVERTISEMENT
Subjects
ADVERTISEMENT
Latest Questions
ADVERTISEMENT