Additive Identity in Linear Algebra: V + 0 = V

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In summary, the conversation discusses the concept of the additive identity and its application in vector spaces. The speaker is trying to prove that the additive identity holds in a problem involving 2x2 matrices and vectors, but there is confusion about the expression and the possibility of it not being a vector space. Testing the axioms is suggested to determine which ones hold.
  • #1
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Hi,

I am new with linear algebra, and I'm having a hard time wrapping my mind around the 0 vector and the additive identity v + 0 = v, where 0 is the 0 vector.
If I had a 2x2 matrix, and v + w = C + (C^T)*D ... (where (C^T) is the transpose, v & w are vectors, and C & D are matrices)... would the additive identity hold? I feel like it wouldn't, because I don't see how it would be unique... but I think I may be wrong.. can someone please help?
 
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  • #2
I can't make sense of your expression, how does adding two vectors get you a 2x2 matrix? Are you confusing something here, or am I the confused one?

Anyway, if the additive identity does not hold, you're not dealing with a vector space and all bets are off (as far as linear algebra is concerned). One of the requirements of a vector space [tex]V[/tex] is that there exists an element [tex]\mathbf{0} \in V[/tex] such that [tex]\mathbf{v} + \mathbf{0} = \mathbf{v}[/tex] for all [tex]\mathbf{v} \in V[/tex].
 
  • #3
That's what I'm trying to prove though, that the additive identity v + 0 = v does in fact hold, and if not it's not a vector space, but we have to test the axioms anyway to see which ones do hold.
Addition of 2 vectors in this problem translates to:
vector v := C (where C is a 2x2 matrix)
vector w:= D (where D is a 2x2 matix)
(v+w):= C + (C^T)D
so i would set up my equation as v + 0 =? v
C + (C^T)D =? C
 
  • #4
If 0 is a 2x2 zero-matrix (the 0-vector you were referring to), then v + 0 = v, but 0 + v = 0, which can't hold for a vector space.
 
  • #5


Hi there,

As a scientist familiar with linear algebra, I can provide some insight into the concept of additive identity in this context. In linear algebra, the additive identity refers to the element that, when added to any other element, results in the same element. This is similar to the concept of 0 in basic arithmetic, where any number added to 0 remains unchanged. In linear algebra, this is represented by the 0 vector, which has all its components equal to 0.

To answer your question, yes, the additive identity holds for any matrix and vector combination, including the one you mentioned (v + w = C + (C^T)*D). This is because the 0 vector, when added to any vector or matrix, will not change its value. In other words, v + 0 = v, and (C + (C^T)*D) + 0 = (C + (C^T)*D), making the additive identity unique and always true.

I hope this helps clarify the concept of additive identity in linear algebra. If you have any further questions, please don't hesitate to ask. Keep exploring and learning about this fascinating field!
 

What is the concept of additive identity in linear algebra?

The concept of additive identity in linear algebra refers to the property that states that when any vector, V, is added to the zero vector, 0, the result will always be the original vector, V. In other words, adding 0 to any vector does not change the vector itself.

How is the additive identity property used in linear algebra?

The additive identity property is used in linear algebra to simplify calculations and proofs. It allows for the manipulation of vectors without changing their values, making it easier to solve equations and prove theorems.

Why is the additive identity important in linear algebra?

The additive identity is important in linear algebra because it is one of the fundamental properties that define vector spaces. It is also used in many linear algebra operations, such as matrix addition and subtraction, and is essential in understanding the concepts of linear independence and spanning in vector spaces.

What happens when a vector is added to its additive identity?

When a vector is added to its additive identity, the result is the same vector. This means that the vector remains unchanged, and the addition has no effect on its value or direction.

Can the additive identity be applied to matrices as well?

Yes, the additive identity property can also be applied to matrices in linear algebra. Just like with vectors, adding a zero matrix to any other matrix will result in the original matrix remaining unchanged.

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