Determine a base and dimension of each of the subspaces below?
#a) S={(x,y,z,w)∈ R^4|x+y-2z=0 and y+3w=0# }
#b) S={A∈M_2 (R) |tr(A)=0} and tr(A)=a_11 +a_22#
1 Answer
a) basis :
b) basis :
Explanation:
a) Since
We can find a basis (and hence settle the question of the dimension) by directly using Gauss-Jordan elimination on the coefficient matrix. In this case, though - the system of constraints are so simple that we can directly write down their implications :
and
So, a general element of the set
Thus all vectors of
is a spanning set. To check whether it is linearly independent we set
which shows that the only combination of
is a linearly independent spanning set - and hence a basis. So, the dimensionality of
b) A general matrix in
where we have used the condition that the matrix is traceless. Thus we can write a general element of
Thus
is a spanning set. It is easy to see that the linear combination