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Linear Independence and Basis

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Questions and Answers

Which of the following statements is true about linear transformations?

Linear transformations can only have one domain.

Linear transformations must be linear maps.

Linear transformations can be nonlinear maps. (correct)

Linear transformations can only have one codomain.

What is the definition of linear independence?

The set of vectors is linearly independent if the equation Ax = b has a unique solution.

The set of vectors is linearly independent if the only solution to the equation Ax = 0 is the trivial solution. (correct)

The set of vectors is linearly independent if the equation Ax = 0 has no solutions.

The set of vectors is linearly independent if the equation Ax = 0 has infinitely many solutions.

What is the range of a linear transformation T?

The set of all possible inputs of T.

The set of all possible solutions to the equation T(x) = 0.

The set of all possible outputs of T. (correct)

The set of all possible solutions to the equation T(x) = b.

What is the definition of a basis?

A basis is a set of vectors that spans the entire vector space. (D) Signup and view all the answers

What is the difference between linear transformations and linear maps?

Linear transformations can be nonlinear maps, while linear maps are always linear. (C) Signup and view all the answers

What is the definition of linear dependence?

Linear dependence is a condition in which a set of vectors can be expressed as a linear combination of other vectors in the same set. Signup and view all the answers

What is the range of a linear transformation with a domain of dimension d?

The range of a linear transformation with a domain of dimension d is a subspace of dimension at most d. Signup and view all the answers

What is the difference between a basis and a linear transformation?

A basis is a set of linearly independent vectors that span a vector space, while a linear transformation is a function that maps vectors from one vector space to another. Signup and view all the answers

What is the codomain of a linear transformation?

The codomain of a linear transformation is the vector space that contains the range of the transformation. Signup and view all the answers

What is the difference between a linear transformation and a linear map?

A linear transformation and a linear map are different terms for the same concept. Both refer to a function that preserves vector addition and scalar multiplication. Signup and view all the answers