- #1

Keesjan

- 5

- 0

## Homework Statement

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

## Homework Equations

## 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?

Last question

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.

Peace