Question

In: Advanced Math

T/F 1) The function f(x) = x1 − x2 + ... + (−1)n+1xn is a linear...

T/F

1) The function f(x) = x1 − x2 + ... + (−1)n+1xn is a linear function, where x = (x1,...,xn).

2) The function f(x1,x2,x3,x4) = (x2,x1,x4,x3) is linear.

3) For a given matrix A and vector b, equation Ax = b always has a solution if A is wide

Expert Solution

1) If and and is scalar, then

Therefore, is linear (TRUE).

2) If and and is scalar, then

Therefore, is linear (TRUE).

3) Consider the wide matrix

and the vector

If has a solution then we get

which means

But the first equation itself implies , so that the system above can not have any solution. Thus, the statement in question is FALSE.

Colby Messinger answered 9 months ago

The production function of a competitive firm is given by f(x1, x2) = x1^1/3 (x2 −...

The production function of a competitive firm is given by f(x1, x2) = x1^1/3 (x2 − 1)^1/3 The prices of inputs are w1 = w2 = 1. (a) Compute the firm’s cost function c(y). (b) Compute the marginal and the average cost functions of the firm. (c) Compute the firm’s supply S(p). What is the smallest price at which the firm will produce? 2 (d) Suppose that in the short run, factor 2 is fixed at ¯x2 = 28. Compute...

T::R2->R2, T(x1,x2) =(x-2y,2y-x). a) verify that this function is linear transformation. b)find the standard matrix for...

T::R2->R2, T(x1,x2) =(x-2y,2y-x). a) verify that this function is linear transformation. b)find the standard matrix for this linear transformation. Determine the ker(T) and the range(T). D) is this linear combo one to one? how about onto? what else could we possibly call it?

3. Given is the function f : Df → R with F(x1, x2, x3) = x...

3. Given is the function f : Df → R with F(x1, x2, x3) = x 2 1 + 2x 2 2 + x 3 3 + x1 x3 − x2 + x2 √ x3 . (a) Determine the gradient of function F at the point x 0 = (x 0 1 , x0 2 , x0 3 ) = (8, 2, 4). (b) Determine the directional derivative of function F at the point x 0 in the direction given...

Let T: R2 -> R2 be a linear transformation defined by T(x1 , x2) = (x1...

Let T: R2 -> R2 be a linear transformation defined by T(x1 , x2) = (x1 + 2x2 , 2x1 + 4x2) a. Find the standard matrix of T. b. Find the ker(T) and nullity (T). c. Is T one-to-one? Explain.

For the given function f(x) = cos(x), let x0 = 0, x1 = 0.25, and x2...

For the given function f(x) = cos(x), let x0 = 0, x1 = 0.25, and x2 = 0.5. Construct interpolation polynomials of degree at most one and at most two to approximate f(0.15)

Suppose that a firm has the p production function f(x1; x2) = sqrt(x1) + x2^2. (a)...

Suppose that a firm has the p production function f(x1; x2) = sqrt(x1) + x2^2. (a) The marginal product of factor 1 (increases, decreases, stays constant) ------------ as the amount of factor 1 increases. The marginal product of factor 2 (increases, decreases, stays constant) ----------- as the amount of factor 2 increases. (b) This production function does not satisfy the definition of increasing returns to scale, constant returns to scale, or decreasing returns to scale. How can this be? (c)Find...

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.)

If the joint probability distribution of X1 and X2 is given by: f(X1, X2) = (X1*X2)/36...

If the joint probability distribution of X1 and X2 is given by: f(X1, X2) = (X1*X2)/36 for X1 = 1, 2, 3 and X2 = 1, 2, 3, find the joint probability distribution of X1*X2 and the joint probability distribution of X1/X2.

1. Let X1, X2 be i.i.d with this distribution: f(x) = 3e cx, x ≥ 0...

1. Let X1, X2 be i.i.d with this distribution: f(x) = 3e cx, x ≥ 0 a. Find the value of c b. Recognize this as a famous distribution that we’ve learned in class. Using your knowledge of this distribution, find the t such that P(X1 > t) = 0.98. c. Let M = max(X1, X2). Find P(M < 10)

Let T(x1, x2) = (-x1 + 3x2, x1 - x2) be a transformation. a) Show that...

Let T(x1, x2) = (-x1 + 3x2, x1 - x2) be a transformation. a) Show that T is invertible. b)Find T inverse.