What is the Basis for the Subspace V in R3 with the Equation x+2y+3z=0?

In summary, the conversation discusses finding a base for a subspace in R3 that has a specific equation and determining if a given space is correct. It is determined that the subspace is a plane and two linearly independent vectors lying in the plane are needed. The given answer is incorrect because it has three basis vectors, and the correct approach is to solve for one unknown and use the other two as parameters. The final basis determined is [(-2, 1, 0), (-3, 0, 1)].
  • #1
gunnar
39
0
Hi. I need to find a base for the subspace V in R3 which has the equation
x+2y+3z=0
Can someone please tell my if the space I'm looking for is

[1 0 0;0 2 0;0 0 3] ?

If not, please explain what I'm doing wrong
 
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  • #2
The subspace is a plane. Find two linearly independent vectors lying in the plane.
 
  • #3
Your answer can't be correct because it has 3 basis vectors and so would span all of R3.
You have one equation is 3 unknowns. Solve for ONE of the unknowns and use the other two as parameters.
x+ 2y+ 3z= 0 so x= -2y- 3z. If you take y= 1, z= 0, then x= -2. One basis vector is (-2, 1, 0). If you take y= 0, z= 1, then x= -3. Another basis vector is (-3, 0, 1).
The basis is [(-2, 1, 0), (-3, 0, 1)].

Of course, a basis is not unique. There are many possible solutions (but they will all contain 2 basis vectors).
 

What is a base for the subspace V in R3?

A base for the subspace V in R3 is a set of linearly independent vectors that span the subspace V. This means that every vector in V can be written as a linear combination of the base vectors.

How do you find a base for the subspace V in R3?

To find a base for the subspace V in R3, you can use the method of Gaussian elimination to reduce the vectors in V to their reduced row-echelon form. The non-zero rows in this form will form a base for V.

Can a subspace V in R3 have more than one base?

Yes, a subspace V in R3 can have infinitely many bases. This is because any set of linearly independent vectors that span V can be considered a base for V.

How many vectors are in a base for the subspace V in R3?

The number of vectors in a base for the subspace V in R3 is known as the dimension of V, denoted as dim(V). It is always a positive integer and represents the minimum number of vectors needed to span V.

Why is it important to find a base for the subspace V in R3?

Having a base for a subspace V in R3 allows us to easily represent any vector in V as a linear combination of the base vectors. This makes it easier to perform calculations and understand the properties of V.

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