Pergunta
3. Determine linear transformation. Find the (4 marks) insformation. bruatrix repressication of 7:R^3arrow R^3 diffined by (T_(1),y,z)=(5x+7y+7y+47z-5z-3y+ relative to the standard (4 marks) 5. Find the equation of the 6. Find the area of a triangle whose vertices are v_(1)(1,2,-1),v_(2)(3,3,1) and 7. Define the linear function f:R^2arrow R^2 by f(x,y)=(x+3y,x+3y) c) Find the basis for the kernel of f and state the nullity of f d) Calculate the rank off (3 marks) (1marks)
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VictorElite · Tutor por 8 anos
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To determine the linear transformation and find its matrix representation, we need to find the images of the standard basis vectors under the transformation T.<br /><br />The standard basis vectors in R^3 are (1,0,0), (0,1,0), and (0,0,1). Applying T to these vectors, we get:<br /><br />T(1,0,0) = 5(1) + 7(0) + 4(0) - 5(0) - 3(0) + 7(0) = 5<br />T(0,1,0) = 5(0) + 7(1) + 4(0) - 5(1) - 3(1) + 7(0) = -1<br />T(0,0,1) = 5(0) + 7(0) + 4(1) - 5(1) - 3(1) + 7(1) = 3<br /><br />Therefore, the matrix representation of T is:<br /><br />[5, 7, 4]<br />[-5, 7, -3]<br />[0, 0, 3]<br /><br />To find the equation of the plane passing through the points (1,2,-1), (3,3,1), and (5,2,-3), we can use the cross product to find a vector perpendicular to the plane. Taking the cross product of the vectors (3,3,1) - (1,2,-1) and (5,2,-3) - (1,2,-1), we get:<br /><br />(2,-4,-8) x (4,-1,-4) = (12, 12, 12)<br /><br />The equation of the plane is then given by the dot product of this vector with the position vector of a point on the plane being zero:<br /><br />12(x - 1) + 12(y - 2) + 12(z + 1) = 0<br /><br />Simplifying, we get:<br /><br />x - y + z + 2 = 0<br /><br />To find the area of the triangle with vertices (1,2,-1), (3,3,1), and (5,2,-3), we can use the formula for the area of a triangle given its vertices:<br /><br />Area = 1/2 * abs(det((v2 - v1), (v3 - v1)))<br /><br />where v1, v2, and v3 are the position vectors of the vertices.<br /><br />Calculating the determinant, we get:<br /><br />det((3-1, 3-2, 1+1), (5-1, 2-2, -3+1)) = det((2, 1, 2), (4, 0, -2)) = 2(0)(-2) - 1(4)(-2) + 2(4)(0) = 8<br /><br />Therefore, the area of the triangle is 1/2 * abs(8) = 4.<br /><br />To find the basis for the kernel of f and state the nullity of f, we need to find the vectors that are mapped to the zero vector under the transformation f.<br /><br />The kernel of f consists of all vectors (x, y) such that f(x, y) = (0, 0). This means that (x + 3y, x + 3y) = (0, 0), which implies that x + 3y = 0. Therefore, the kernel of f consists of all vectors of the form y(-3, 1).<br /><br />The basis for the kernel of f is therefore (-3, 1), and the nullity of f is 1.<br /><br />To calculate the rank of f, we need to find the dimension of the image of f. Since the image of f is the set of all vectors of the form (x + 3y, x + 3y), the rank of f is 1.
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