Solve Matrix Hyperplane: Find A & B | Please Help!

[Mathguy]

Please Help!

I can't figure this out no matter how many times I try. Please can someone help me with this homework question?:

Let:

v1=[1]
[1]
[0]
[0]

v2=[1]
[0]
[1]
[0]

v3=[1]
[0]
[0]
[1]

v4=[1]
[1]
[1]
[1]

and let
V= Span{v1,v2,v3) intersection Span{v4}^perp

(sorry i don't know how to type the symbols for intersection and perpenducular)

In other words, V is the set of vectors x in the hyperplane Span{v1,v2,v3) which also satisfy the equation v4 · x = 0, i.e. x1+x2+x3+x4=0.

a) Find a matrix A such that V=N(A)
b) Find a matrix B such that V=C(B)

Thanks in advance!

 

[so-crates]

What do V = N(A) and V = C(B) mean ?

 

[Mathguy]

N(A) is the nullspace of A.
C(B) is the column space of B.

The question asks to find a matrix A such that the vector space V equals the nullspace of A, and to find a matrix B such that the vector space V equals the column space of B.

 

[so-crates]

I think you got your terminology a bit mixed up, they want you to find matrices A and B such that V *spans* the nullspace of A and V *spans* the nullspace of in other words find a matrix such that

AV = 0 where is the 0 vector [0 0 0 0] (vertically though)

BV = span(V)

Let me given you a hint: what is the span of the matrix consisting of the vectors [v1 v2 v3 v4] ? Do the vectors happen to be linearly independent? If so what would their span be ? If not, what set of vectors forms a linearly independent set ? Now that you know the span of the set, can't you find its kernel? What would that mean about the nullspace?