K^4/Example for intersection and sum/Bases/2/Exercise

Let ${}K$ be a field.

a) Show that the linear subspace ${}U\subseteq K^{4}$ generated by

${\begin{pmatrix}5\\3\\-1\\4\end{pmatrix}},\,{\begin{pmatrix}2\\6\\5\\-3\end{pmatrix}},\,{\begin{pmatrix}4\\-2\\-1\\1\end{pmatrix}}$

has dimension ${}3$.

b) Determine a basis and the dimension of the solution space ${}L\subseteq K^{4}$ of the linear equation

${}7x_{1}+5x_{2}+3x_{3}-6x_{4}=0\,.$

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

d) Confirm fact in this example.

Create a solution