Proving W is a Subspace of V: Let u & v be Vectors in V

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Discussion Overview

The discussion revolves around proving that the set W of all linear combinations of two fixed vectors u and v in a vector space V is a subspace of V. The focus is on the proof structure and the necessary steps to demonstrate this property.

Discussion Character

  • Homework-related

Main Points Raised

  • Some participants express difficulty in proving that W is a subspace and seek assistance.
  • One participant suggests taking two vectors from W and showing that their linear combination is also in W.
  • A later reply clarifies that "taking two vectors from W" refers to taking two linear combinations of u and v, and emphasizes the need to demonstrate that their linear combination can also be expressed in the form of a linear combination of u and v.

Areas of Agreement / Disagreement

Participants generally agree on the approach to proving that W is a subspace, but the initial confusion indicates that not all aspects of the proof are clear to everyone.

Contextual Notes

The discussion does not resolve all assumptions or steps necessary for the proof, leaving some details about the proof structure and definitions potentially unclear.

ECE
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Let u and v be (fixed) vectors in the vector space V. Show that the set W of all linear combinations au+bv of u and v is a subspace of V.

I cannot prove the above proof properly. Can anyone help.

-Thanks
 
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ECE said:
Let u and v be (fixed) vectors in the vector space V. Show that the set W of all linear combinations au+bv of u and v is a subspace of V.

I cannot prove the above proof properly. Can anyone help.

-Thanks

Take two vectors from W, and show that their linear combination is also in W.
 
"Take two vectors from W" means taking two linear combinations, perhaps with different "a" and "b', say au+ bv and cu+ dv. "Their linear combination" would be something like x(au+ bv)+ y(cu+ dv) for numbers, x and y. "Show it is also in W" means "show it satisfies the definition of W". Here that means show that it also can be written au+ bv for some choices of a and b.
 
Thanks i understand it now
 

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