Find a linear transformation that maps the point (0, 0) to itself, the point (0,

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Find a linear transformation that maps the point (0, 0) to itself, the point (0, 1) to (-1, 1) and the point (1, 0) to (0, -10). Find a linear transformation that maps every point on the line y = 3x to a point on the line y = x If we apply the previous linear transformations in the previous two problems to the unit square, what is the result? Compute the determinants of the matrices associated to the linear transformations in the previous problems. Show that the transformation f rightarrow integral_0^x f(s) ds defined as a map from the vector space C([0, 1]) of all continuous functions defined in the interval [0, 1] to itself, is linear. Show that the transformation f rightarrow f' defined as a map from the vector space C^1([0, 1]) of all functions with continuous first derivatives defined in the interval [0, 1] to the vector space C([0, 1]), is linear. Verify that the following transformation (x_1, x_2, x_3) rightarrow (y_1, y_2, y_3) is linear and find the matrix associated to the transformation y_1 = x_1 - 2x_2 + 10x_3 y_2 = -x_3 + x_1 y_3 = -x_2. A solid Q in space is obtained by applying the transformation in the previous problem to the unit cube {{x, y, z) such that 0 lessthanorequalto x lessthanorequalto y lessthanorequalto 1 and 0 lessthanorequalto z lessthanorequalto 1}. Find the volume of Q. Eigenvalues and Eigenvectors Definition 6.1. Let A be an n times n matrix. If lambda C and upsilon elementof R^n {0} satisfy A upsilon = lambda upsilon, then A is called an eigenvalue of A with associated eigenvector upsilon. If A upsilon = lambda upsilon, then (A - lambda I) upsilon = 0. Since upsilon is nonzero, A - lambda I is a singular matrix d, and we know

Explanation / Answer

3) We have to find a linear transformation T such that T(0,0) = (0,0), T(0,1) = (-1,1), T(1,0) = (0,-10).

Any vector (x,y) can be written as (x,y) = x(1,0) + y(0,1)

Thus T(x,y) = x T(1,0) + y T(0,1)

or T(x,y) = x (0,-10) + y (-1,1)

or T(x,y) = (0-y, -10x+y) = ( -y, y-10x )

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