Let V be the vector space of all 2 x 2 matrix over the field R of real numbers and B = . If T: V-- Vis a linear transformation defined by T(A) = AB - BA, then what is the dimension of the Kernel of T?a)1b)2c)3d)4Correct answer is option 'B'. Can you explain this answer?

Question

Let V be the vector space of all 2 x 2 matrix over the field R of real numbers and B = . If T: V--> Vis a linear transformation defined by T(A) = AB - BA, then what is the dimension of the Kernel of T?a)1b)2c)3d)4Correct answer is option 'B'. Can you explain this answer?

Answer

Let V be the vector space of all 2 x 2 matrices over the field R of real numbers and matrix

Let T : V —> V be a linear transformation defined by T(A) = AB - BA
We need to determine the dim of ker T.
Let

Then, take T(A) = 0 implies AB - BA = 0
Substituting the values of matrices A and B,
we get

or equivalently

=

or equivalently

implies c =0 and a + b - d = 0 Therefore,

Hence, dim (ker T) = Total number of variables - Number of restrictions = 4 - 2 = 2

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