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)

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.