Subspace in R^4: Investigating (2x+3y, x, 0, 1) as a Potential Subspace

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SUMMARY

The discussion focuses on determining whether the set defined by the expression (2x+3y, x, 0, 1) constitutes a subspace of R^4. Key criteria for a subset to qualify as a subspace include the existence of an additive identity, closure under vector addition, and closure under scalar multiplication. The analysis reveals that the set fails to meet these criteria, particularly due to the constant term '1' in the fourth component, which prevents the existence of an additive identity and violates closure under scalar multiplication.

PREREQUISITES
  • Understanding of vector spaces and their properties
  • Knowledge of R^4 and its components
  • Familiarity with the concepts of additive identity and closure properties
  • Basic algebraic manipulation skills
NEXT STEPS
  • Review the definition of vector spaces and subspaces in linear algebra
  • Study examples of subspaces in R^n, focusing on R^4
  • Learn about the implications of constant terms in vector expressions
  • Explore closure properties in vector addition and scalar multiplication
USEFUL FOR

Students studying linear algebra, particularly those grappling with concepts of vector spaces and subspaces in R^4, as well as educators seeking to clarify these foundational topics.

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Homework Statement



Is this a subspace of R^4, (2x+3y, x, 0 , 1) . Give reasons

Homework Equations





The Attempt at a Solution



I am completely stuck at this one
 
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A subset U of V is a subspace of V if it satisfies the properties needed to be a vector space: additive identity; closure under vector addition; closure under scalar multiplication.

Check that your given subset satisfies the properties.

a) Is there an additive identity from your set for R^4?

b) If you take two vectors from your given space, a and b, is a+b still in your set?

c) Is the scalar multiple of any vector in your set still in the set?
 
Last edited:
You might want to look particularly at the fourth component in a scalar product such as 2v.
 

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