Lin. transf. and linear independence

Thread starter eckiller
Start date Feb 9, 2005
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Independence Linear Linear independence

In summary, a linear transformation is a function that preserves the structure of a vector space. To determine if a linear transformation is invertible, it must be both injective and surjective. A set of vectors is linearly independent if no vector can be created by combining the others. To test for linear independence, a system of equations can be set up and checked for a non-trivial solution. It is possible for a set of vectors to be linearly independent in one vector space but not in another, as the definition of linear independence depends on the underlying vector space and its operations.

Feb 9, 2005

eckiller

Hi,

If I transform a set of linearly independent vectors by a one-to-one linear transformation, is the transformed set also linearly independent?

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Feb 9, 2005

Hurkyl

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Suppose it isn't. What can you say about it?

Feb 9, 2005

mathwonk

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make sure you know all the definitions of the terms you are using. deciding this problem comes almost immediately just from knowing what the words mean.

1. What is a linear transformation?

A linear transformation is a function that maps one vector space to another in a way that preserves the underlying structure of the vector space, such as addition and scalar multiplication.

2. How do you determine if a linear transformation is invertible?

A linear transformation is invertible if and only if it is both injective (or one-to-one) and surjective (or onto). This means that every input has a unique output and every output has at least one input.

3. What does it mean for a set of vectors to be linearly independent?

A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the other vectors. In other words, none of the vectors can be "created" by combining the others.

4. How do you test for linear independence?

To test for linear independence, you can set up a system of equations using the vectors in the set and see if there is a non-trivial solution (a solution other than all variables being equal to zero). If there is no non-trivial solution, then the vectors are linearly independent.

5. Can a set of vectors be linearly independent in one vector space but not in another?

Yes, a set of vectors can be linearly independent in one vector space but not in another. This is because the definition of linear independence depends on the underlying vector space and the operations defined within it. So, a set of vectors may be linearly independent in one vector space but not independent in another vector space with different operations.