Explain whether T (R^2 -R) is a linear transformation. If it is a linear transformation, supply proof, and if it is not, supply a counterexample: T(a,b) = a + b
Let x = (a, b) and y = (\alpha, \Beta) all elements of R^2, and k,a,b,\aplha,\Beta all elements of R. Thus, T(kx) = T(ka,kb) = (ka) + (kb) = k(a + b) = kT(a,b) = kT(x) T(x + y) = T(a + \alpha, b + \Beta) = (a + \alpha) + (b + \Beta) = (a + b) + (\alpha +\Beta) = T(a, b) + T(\alpha, \Beta) = T(x) + T(y) Thus, From the above proof, we can confirm that T is Linear.
State whether the following question is true or false: A qualitative variable that categorises or describes or names an element of a population is referred to as a normal variable.
Can Cramer's Rule be applied on the following system of Linear equations: x1.cos(y) - x2.sin(y) = 1 x1.sin(y) + x2.cos(y) = -3
We first determine the determinant of the coefficient matrix. The determinant is: | cos(y) -sin(y) | | sin(y) cos(y) | =cos^2 (y) + sin^2 (y) = 1 Thus, the system of equations is not equal to 0. Therefore the non-homogeneous and we can conclude that Cramer's Rule can be applied to the linear equations.