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

The independent-column representation gives $T(x)=c_1(x_1+3x_3)+c_2(x_2+2x_4)$. Thus a vector lies in the kernel of a linear map represented by this matrix precisely when $x_1=-3x_3$ and $x_2=-2x_4$. Writing the two free coordinates separately givesThe two displayed vectors are linearly independent, so they form a basis of the kernel. Its dimension two also agrees with the rank-nullity theorem, $4=2+2$.