Vector Addition Properties and Definitions

The video tutorial explores a redefined addition operation for vectors in R2, where the sum of two vectors is calculated differently. It examines whether this new operation satisfies key vector space axioms, such as the existence of a zero vector and the commutative property. The tutorial confirms the presence of a zero vector but finds that the commutative property does not hold, concluding that the set is not a vector space under the new addition definition.

What is the new definition of vector addition in R2?

• a + b, c + d

• a + b, c + d

• a + d, b + c

• a + c, b + d

What is the usual way of defining multiplication in vector spaces?

• Redefined as addition

• Defined as the usual way

• Defined as division

• Defined as subtraction

What is the zero vector in the context of the new addition operation?

• A vector with negative entries

• A vector with one entry as zero

• A vector with both entries as zero

• A vector with all entries as one

Does the sum of two vectors under the new addition remain in R2?

• Yes, but only if vectors are identical

• No, it moves to R3

• Yes, it remains in R2

• No, it becomes a scalar

What is the first entry of the sum of vectors u and v under the new addition?

• v1 + u2

• u2 + v2

• u1 + v2

• u1 + v1

Is the new vector addition commutative?

• Depends on the scalar multiplication

• No, it is not commutative

• Yes, it is commutative

• Only for certain vectors

What is the second entry of the sum of vectors v and u under the new addition?

• u2 + v1

• v1 + u2

• u1 + v2

• v2 + u1

What is the result of u1 + v2 compared to v1 + u2 under the new addition?

• They are equal

• They are not equal

• u1 + v2 is greater

• v1 + u2 is greater

Why does the new addition operation fail to form a vector space?

• It lacks a zero vector

• It is not closed under addition

• It fails the commutative property

• It does not allow scalar multiplication

What is the conclusion about the vector space with the new addition?

• It forms a vector space only with specific scalars

• It forms a vector space only with specific vectors

• It does not form a vector space

• It forms a vector space