Well as the title describes, x and y linear independent in R^n S is a subspace in Rn spanned by x,y i.e S= span(x,y)
define the matrix A as A=xy^T+yx^T
This is actually a 3 part question
1 show that A is symmetrical
2 show that N(A)=S(Perpendicular)
3 show that the rank of A is 2
The Attempt at a Solution
I think i got the first one
A^T= (xy^T+yx^T)^T = (y^T)^Tx^T+(x^T)^Ty^T= yx^T+xy^T=A QED
Dimension is abbreviated as Dim
Rowspace abbreviated as R(*)
Column space will be abbreviated as C(*)
and perpendicular as perp
Now the second one i think has something to do with the N(A)=C(A^T)^perp.
And because A^T=A now N(A)=C(A)^perp
Span of the DimC(A) is span of S because x,y are linear independent?
And if DimC(A)=Span(x,y) then DimC(A)=S
So filling it all in N(A)=S^perp QED??
is this correct?
Since x and y are linear independent they form the basis of A. So DimC(A)=2
And since the DimC(A)=DimR(A) and the rank=DimR so Rank(A)=2
Any feed back would be nice.