Is W a Subspace of R^3 and What is Its Geometry?

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SUMMARY

The discussion centers on determining whether the set W defined by W={x:x3=2x1-x2} is a subspace of R^3. It is established that W is indeed a subspace, as it satisfies the properties of closure under addition and scalar multiplication. The geometric description of W reveals that it represents a plane in R^3, specifically defined by the equation 2x1 - x2 - x3 = 0. The basis for this subspace is given by the vectors [1, 0, 2] and [0, 1, -1].

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  • Understanding of vector spaces and subspaces in linear algebra
  • Familiarity with R^3 and its geometric interpretations
  • Knowledge of linear combinations and basis vectors
  • Ability to work with equations of planes in three-dimensional space
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  • Study the properties of vector spaces and subspaces in linear algebra
  • Learn how to derive the equation of a plane from a set of vectors
  • Explore the concept of basis and dimension in vector spaces
  • Investigate linear transformations and their geometric implications
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Students and educators in mathematics, particularly those focusing on linear algebra, as well as anyone interested in understanding the geometric properties of vector spaces in R^3.

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I'm stuck on a problem which asks:
Determine whether W is a subspace of R^3. If W is a subspace, then give geometric description of W. The problem is W={x:x3=2x1-x2} and x=[x1, x2, x3]
I tried solving it but I'm having a hard time understanding the properties of R^n and using them. I guess I'm not as good with proof as I'm with numbers. Please help.
 
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Because x3= 2x1- x2, any vector in that subspace can be written as [x1, x2, x3]= [x1, x2, 2x1- x2]= [x1, 0, 2x1]+ [0, x2, -x2]= x1[1, 0, 2]+ x2[0, 1, -1].

Can you see what a basis for that subspace is? Of course, 2x1- x2- x3= 0 is the equation of a plane.
 

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