Let M22 be the vector space of 2x2 matrices and let TM22 R b

Let M₂₂ be the vector space of 2x2 matrices, and let T:M₂₂ ->R be defined by T(A) = a+d ... a) Show that T is linear Transformation b) Describe the Kernel of T and determine a basis for Ker T...

Solution

If we take an assumption of R being a non vector , we can jump to a conclusion where we cannot prove T to be a linear transformation as \" a+d\" is not possible. We can\'t do addtion operation on a vector and a scalar quantity.

On second assumption ( of R being a scalar) also , we cannot show T as linear transformation as addition rule is not preserved as shown below.

T(A) = a + d

T(C) = c + d

T(A+C) must be equal to T(A) + T(C) for T to being a linear transformation.

But, we have T(A+C) = a + c + d and T(A) + T(C) = a +d +c + d = a + c + 2d

Since, T(A+C) is not equal to T(A) + T(C) , addition rule is not preserved and therefore it is not a linear transformation .