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?

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

What is the difference between linear transformations and linear maps?

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

What is the definition of linear dependence?

<p>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.</p> Signup and view all the answers

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

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

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

<p>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.</p> Signup and view all the answers

What is the codomain of a linear transformation?

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

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

<p>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.</p> Signup and view all the answers

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