Question regarding orthogonality and linear independence

Question

I am new to linear algebra and was a bit confused regarding the following… Any feedback would be really appreciated...

True or false?

1. If $K$ is a non-empty set of vectors in $R^n$, then $(K^\bot)^\bot=K$.
2. $A=\{v_1,v_2\}$ is a set of vectors in $R^4$. If $(Sp(A))^\bot=Sp\{(1,2,1,1),(2,2,2,2),(2,1,2,2)\}$, then A is linearly dependent.

My answer for 1 is true, per definition.

My answer for 2 is:

The vectors in $Sp\{(1,2,1,1),(2,2,2,2),(2,1,2,2)\}$ are linearly dependent, and can be reduced to $Sp\{(1,2,1,1),(2,2,2,2)\}$. This means that the dimension of $Sp(A)^\bot$ is 2 and the same for $Sp(A)$. Based on $A^\bot=Sp\{(1,2,1,1),(2,2,2,2)\}$, $(A^\bot)^\bot=Sp\{(-1,0,1,0),(-1,0,0,1)$, which I worked out via a matrix based on $(x,y,z,w)·(1,2,1,1)=0$ and $(x,y,z,w)·(2,2,2,2)=0$. The vectors in $(A^\bot)^\bot=Sp\{(-1,0,1,0),(-1,0,0,1)$ are linearly independent, so the statement is incorrect.

Thank you!

Answer 1

1. Be careful, $(K^{\perp})^{\perp}$ is always a subspace. So what if $K$ is just a set and not a subspace?
2. You reasoning seems correct.