Linear algebra please provide a clear and detailed reponse t

Linear algebra, please provide a clear and detailed reponse to the following, thanks:

Let v = (2 1) and w = (1 3). Are v, w dependent or independent? What is the span of v and w (geometrically, in words)? Does there exist a third nonzero vector t = (t_1 t_2) that is simultaneously perpendicular to both v and w? If so, find t.

Solution

v and w are independent

Proof

Let, a,b so that

av+bw=0

2a+b=0

a+3b=0

Solving gives,

a=b=0

v and w are of size 2x1

Hence ,v and w span R2 ie whole of 2 dimensional space, R2

No there does not exist such a vector t because any vector of size 2x1 is in span of v and w.