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

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In summary, the conversation is about finding matrices A and B such that the vector space V spans the nullspace of A and the column space of B. The vectors v1, v2, v3, and v4 are given and the question asks to find a matrix A such that V=N(A) and a matrix B such that V=C(B). It also provides the definitions of N(A) and C(B) as the nullspace and column space of A and B, respectively. The conversation also includes hints on how to approach the problem, such as finding the span and kernel of the given vectors.
  • #1
Mathguy
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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!
 
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  • #2
What do V = N(A) and V = C(B) mean ?
 
  • #3
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.
 
  • #4
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?
 
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FAQ: Solve Matrix Hyperplane: Find A & B | Please Help!

How do you solve a matrix hyperplane?

To solve a matrix hyperplane, you will need to find the coefficients A and B that satisfy the equation Ax + By = 0, where x and y are variables. This can be done by using various methods such as Gaussian elimination, Cramer's rule, or matrix inversion.

What is the importance of solving a matrix hyperplane?

Solving a matrix hyperplane is important in linear algebra and other mathematical applications. It can help determine the relationship between two variables and can be used to solve systems of equations.

Are there any specific steps to solve a matrix hyperplane?

Yes, there are specific steps to solve a matrix hyperplane. These steps may vary depending on the method chosen, but generally involve setting up an augmented matrix, row reducing to obtain a triangular form, and then solving for the variables.

Can a matrix hyperplane have more than two variables?

Yes, a matrix hyperplane can have more than two variables. The number of variables in a hyperplane is determined by the number of rows in the matrix. A hyperplane with n variables will have n equations and n unknowns.

How do I know if my solution to a matrix hyperplane is correct?

You can check the correctness of your solution by plugging in the values of A and B into the original equation and ensuring that both sides are equal. You can also use a graphing calculator to plot the hyperplane and see if it passes through the given points.

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