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brushman
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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.
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 ∅.
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.
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.
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.
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.