# Find cartesian equation of hyperplane spanned by a set of vectors

• Mr. Johnson
In summary, Mr. Johnson found the Cartesian equation for the hyperplane W in R4 spanned by the column vectors v1, v2, and v3.
Mr. Johnson
Let W be a hyperplane in R4 spanned by the column vectors v1 , v2, and v3, where

Note that these are suppose to be COLUMN vectors:

v1 = [3,1, -2 , -1], v2 = [0, -1, 0 , 1], v3= [1,2 ,6, -2]

Find the Cartesian (i.e., linear) equation for W.

I'm not quite sure where to start or how to interpret this problem. I was thinking about first finding the span or column space?... But after that I would not know how to convert into a cartesian equation.

Any guidance or tips would be appreciated.

Thank you.

Good morning, Mr. Johnson, how are you!

Hint: how would you find the Cartesian equation of the plane in R3 spanned by two column vectors v1 and v2 ?

Ok so I found an example but I feel like it isn't very intuitive and that there are better methods and approaches to this problem.

So here is what I did:

The hyperplane W is of the form Ax1 + Bx2 + Cx3 + Dx4 = 0 since it must pass through the origin.

rref ( 3 1 -2 -1) = [1 0 0 0]
(0 -1 0 1 ) [ 0 1 0 -1]
(1 2 6 -2 ) [0 0 1 0]

Thus A = C =0
B = D

=> 0x1 + Bx2 +0x3 + Bx4 =0

Dividing by B => x2 + x4 =0 which is the final answer for the cartesian equation for W.

Can anyone verify? I know there is a better approach to this problem.

not actually following what you've done there ,

but your result x2 + x4 = 0 is obviously correct!

the more general method i was thinking of (for two vectors in ℝ3) was to find their cross product using a determinant …

can you see a 4D equivalent of that?

A little more direct way: Any vector in the span of those three vectors can be written
<x, y, z, t>= a<3, 1, -2, -1>+ b<0, -1, 0, 1>+ c<1, 2, 6, -2>= <3a+ c, a- b+ 2c, -2a+ 6c, -a+ b- 2c>

so that x= 3a+ c, y= a- b+ 2c, z= -2a+ 6c, t= -a+ b- 2c. Those are parametric equations for the plane, also there are three equations in the three "unknowns", a, b, and c. Can you solve the equations for a, b, and c, to get just a single equation in x, y, z.

## 1. What is a hyperplane?

A hyperplane is a geometric object in n-dimensional space that is one dimension less than the space it is in. In other words, in a three-dimensional space, a hyperplane is a two-dimensional object, such as a flat plane. In a two-dimensional space, a hyperplane is a one-dimensional object, such as a straight line.

## 2. What does it mean for a hyperplane to be spanned by a set of vectors?

When a hyperplane is spanned by a set of vectors, it means that the hyperplane contains all linear combinations of those vectors. In other words, any point on the hyperplane can be expressed as a linear combination of the set of vectors.

## 3. How do you find the cartesian equation of a hyperplane spanned by a set of vectors?

To find the cartesian equation of a hyperplane spanned by a set of vectors, you first need to find a basis for the hyperplane. This can be done by solving a system of linear equations using the set of vectors as the coefficients. Once you have the basis, you can use it to write the equation of the hyperplane in the form Ax + By + Cz + ... = D, where A, B, C, etc. are constants.

## 4. Can a hyperplane be spanned by more than one set of vectors?

Yes, a hyperplane can be spanned by more than one set of vectors. In fact, there are infinitely many sets of vectors that can span the same hyperplane. However, the basis and cartesian equation of the hyperplane will be different for each set of vectors.

## 5. Are there any applications of finding the cartesian equation of a hyperplane?

Yes, there are many applications of finding the cartesian equation of a hyperplane. Some examples include using it in linear regression models, solving systems of linear equations, and finding the equation of a plane in 3D space. It is also a fundamental concept in linear algebra and is used in various fields such as physics, engineering, and computer graphics.

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