K^3/Example for intersection and sum/Bases/1/Exercise

Let ${}K$ be a field.

a) Show that the linear subspace ${}U\subseteq K^{3}$ generated by

${\begin{pmatrix}3\\2\\-4\end{pmatrix}},\,{\begin{pmatrix}2\\8\\-3\end{pmatrix}}$

has the dimension ${}2$.

b) Determine a basis and the dimension of the solution space ${}L\subseteq K^{3}$ of the linear equation

${}-6x_{1}+4x_{2}+5x_{3}=0\,.$

c) Determine a basis and the dimension of the intersection ${}U\cap L$.

d) Confirm fact in this example.

Create a solution