 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/digital-systems-linear-algebra-introduction-to-formal-proof-resit/4/b/ii/solution

The independent-column representation gives

T (x) = c_{1} (x_{1} + 3 x_{3}) + c_{2} (x_{2} + 2 x_{4})

. Thus a vector lies in the kernel of a linear map represented by this matrix precisely when

x_{1} = - 3 x_{3}

and

x_{2} = - 2 x_{4}

. Writing the two free coordinates separately gives

x = x_{3} - 3 010 + x_{4} 0 - 2 01 .

The two displayed vectors are linearly independent, so they form a basis of the kernel. Its dimension two also agrees with the rank-nullity theorem,

4 = 2 + 2

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