Question on Remark in Axler's Linear Algebra

The example given by Axler is not a subspace unless b=0 due to the requirement that the zero vector must be contained within the subspace. This is the only solution that allows for closure under addition and scalar multiplication.
  • #1
Group_Complex
81
0
Hello, i am studying vector subspacess and Axler introduces the two criteria for a vector subspace (closure under addition and scalar multiplication).
He then proceeds to give an example; (x1,x2,x3,x4) belonging to F^4 : x3=x4+b, where b is an element of F. Axler states that this example is not a subspace unless b=0, yet this is the same space as V and i was under the impression (Axler states it himself) that V is a subspace of itself? Should not any value of b in F be possible?
 
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  • #2
I think i have realized where i went wrong. The zero vector must be contained within the subspace, thus b=0 is the only solution which allows this. Is that a suitable method to complete the example or am i missing something else?
 
  • #3
Group_Complex said:
I think i have realized where i went wrong. The zero vector must be contained within the subspace, thus b=0 is the only solution which allows this. Is that a suitable method to complete the example or am i missing something else?

that is correct.
 

What is the significance of the remark in Axler's Linear Algebra?

The remark in Axler's Linear Algebra is significant because it highlights the importance of understanding the difference between finite-dimensional and infinite-dimensional vector spaces. It also emphasizes the fact that certain theorems and concepts may hold true for finite-dimensional spaces but not for infinite-dimensional spaces.

Why does Axler make this remark in his textbook?

Axler makes this remark in his textbook to caution readers about the limitations of certain theorems and definitions in the context of infinite-dimensional vector spaces. It serves as a reminder to approach linear algebra with a critical mindset and to be aware of the assumptions and generalizations being made.

Can the remark in Axler's Linear Algebra be applied to other areas of mathematics?

Yes, the remark in Axler's Linear Algebra can be applied to other areas of mathematics such as functional analysis and abstract algebra. Understanding the differences between finite and infinite structures is crucial in many mathematical fields and this remark serves as a reminder of that.

Do all mathematicians agree with Axler's remark?

No, not all mathematicians may agree with Axler's remark. Some may argue that certain theorems and concepts can be extended to infinite-dimensional spaces, while others may argue for the importance of studying both finite and infinite-dimensional spaces separately. However, the remark serves as a valuable perspective and reminder for mathematicians to critically evaluate the applicability of linear algebra concepts.

How can the remark in Axler's Linear Algebra be practically applied in research or real-world problems?

The remark in Axler's Linear Algebra can be practically applied in research and real-world problems by guiding mathematicians to carefully consider the assumptions and limitations of their methods and results. It also encourages the exploration of alternative approaches and solutions in the context of infinite-dimensional spaces.

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