Does the 0 vector not count when determining the dimension?

In summary, the 0 vector is a vector with all of its components equal to zero, often denoted as 0 or ∅. The dimension of a vector space is determined by the number of linearly independent vectors that span the space, and the 0 vector does not count towards this dimension since it is a linear combination of itself. A vector space can have a dimension of 0, meaning it only contains the 0 vector and is a single point in space. Understanding the role of the 0 vector in determining dimension is important for accurately describing and analyzing vector spaces, distinguishing between different vector spaces, and applying this concept in various fields.
  • #1
brushman
113
1

Homework Statement


If we have ker(T)={0}, why is the dim(ker(T)) = 0?

Does the 0 vector not count when determining the dimension? I thought the answer would be 1.
 
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  • #2
Nope, doesn't count. The zero vector is in any vector space. Even a one-dimensional space has the zero vector in it. The zero vector isn't what makes it one-dimensional.
 

1. What is the 0 vector?

The 0 vector, also known as the zero vector or null vector, is a mathematical construct that represents a vector with all of its components equal to zero. It is often denoted as 0 or .

2. How is the dimension of a vector space determined?

The dimension of a vector space is determined by the number of linearly independent vectors that span the space. In other words, it is the minimum number of vectors required to express any vector in the space as a linear combination of those vectors.

3. Does the 0 vector count towards the dimension of a vector space?

No, the 0 vector does not count towards the dimension of a vector space. This is because the 0 vector is a linear combination of itself, and therefore it does not add any new information or contribute to the span of the vector space.

4. Can a vector space have a dimension of 0?

Yes, a vector space can have a dimension of 0. This means that the space only contains the 0 vector, and any other vector can be expressed as a linear combination of the 0 vector. In other words, the space is a single point in space.

5. Why is it important to understand the role of the 0 vector in determining dimension?

Understanding the role of the 0 vector in determining dimension is important because it helps us accurately describe and analyze vector spaces. It also allows us to distinguish between different vector spaces and understand their properties and relationships. Additionally, it is a fundamental concept in linear algebra and is used in various applications in science and engineering.

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