Proving Vector Space of U is Null Space of T

Thread starter franky2727
Start date Jan 3, 2009
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Null space Space

In summary, a vector space is a mathematical structure that consists of a set of elements and operations that satisfy certain properties. The null space of a linear transformation or matrix is the set of all vectors mapped to the zero vector, and for a vector space to be the null space of a transformation, every vector in that space must be mapped to the zero vector. To prove that a vector space is the null space of a transformation, one must show that the range of the transformation is the zero vector and that the null space is equal to the vector space. This is important because it allows for a better understanding of the properties and connections between the two structures, and is crucial in solving linear systems and understanding linear transformations.

Jan 3, 2009

franky2727

got a question show that the null space of T is a vector space of U given the mapping T:U->V

i know that null space or kernal of T is kerT={uEU: T(u)=0} and is a subset of U but don't have a clue where to start applying this to my question?

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Jan 3, 2009

radou

Homework Helper

Well, how do you show that a subset of U is a subspace of U?

1. What is a vector space?

A vector space is a mathematical structure that consists of a set of elements (vectors) and operations (such as addition and scalar multiplication) that satisfy certain properties. These properties include closure, associativity, commutativity, existence of an identity element, and existence of inverse elements.

2. What is a null space?

In linear algebra, the null space of a linear transformation or matrix is the set of all vectors that are mapped to the zero vector by the transformation or matrix. In other words, the null space is the set of all solutions to the homogeneous equation Ax = 0, where A is a matrix and x is a vector.

3. What does it mean for a vector space to be the null space of a transformation?

If a vector space U is the null space of a linear transformation T, it means that every vector in U is mapped to the zero vector by T. In other words, the range of T is the zero vector, and the null space of T is the entire vector space U.

4. How do you prove that a vector space U is the null space of a transformation T?

To prove that a vector space U is the null space of a transformation T, you must show that every vector in U is mapped to the zero vector by T. This can be done by showing that the range of T is the zero vector and that the null space of T is equal to U.

5. Why is it important to prove that a vector space U is the null space of a transformation T?

Proving that a vector space U is the null space of a transformation T is important because it allows us to understand the properties of both U and T. It also allows us to make connections between the two structures and use the properties of one to understand the properties of the other. Additionally, proving that a vector space is the null space of a transformation is an important step in solving linear systems and understanding the behavior of linear transformations.