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Questions and Answers
What is a basis of a subspace?
What is a basis of a subspace?
What does the dimension of a subspace represent?
What does the dimension of a subspace represent?
The number of vectors in any basis of the subspace.
The pivot columns of a matrix A form a basis for the column space of A.
The pivot columns of a matrix A form a basis for the column space of A.
True
The free variables in the solution set of Ax = 0 do not contribute to the basis of the null space Nul(A).
The free variables in the solution set of Ax = 0 do not contribute to the basis of the null space Nul(A).
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What is the Rank Theorem?
What is the Rank Theorem?
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What does the Basis Theorem state?
What does the Basis Theorem state?
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How can a matrix be viewed in the context of functions?
How can a matrix be viewed in the context of functions?
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What is an identity transformation?
What is an identity transformation?
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What does the matrix transformation associated with a matrix A do?
What does the matrix transformation associated with a matrix A do?
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The range of a matrix transformation is always the entire space Rm.
The range of a matrix transformation is always the entire space Rm.
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Study Notes
Basis of a Subspace
- A basis of a subspace V in Rⁿ consists of vectors {v₁, v₂,..., vm} that both span V and are linearly independent.
Dimension
- The dimension of a subspace V, denoted as dimV, is determined by the number of vectors in any basis of V.
A Basis for the Column Space
- The pivot columns of a matrix A serve as a basis for its column space, Col(A).
A Basis for the Null Space
- The vectors associated with free variables in the solution of Ax = 0 form a basis for the null space, Nul(A).
The Rank Theorem
- The rank of matrix A, noted as rank(A), is the dimension of Col(A).
- The nullity of matrix A, nullity(A), is the dimension of Nul(A).
- Relationships: rank(A) = dim Col(A) (number of pivot columns) and nullity(A) = dim Nul(A) (number of free variables).
- For an m x n matrix A: rank(A) + nullity(A) = n.
The Basis Theorem
- For a subspace V of dimension m:
- Any set of m linearly independent vectors in V is a basis for V.
- Any set of m vectors that spans V is also a basis for V.
Matrices as Functions
- A matrix A with m rows and n columns can be viewed as a function mapping input vector x (in Rⁿ) to output vector b (in Rm) through the equation b = Ax.
- The set of all possible output vectors from this transformation forms the range dictated by the solutions of Ax = b.
Transformation
- A transformation (or map) T assigns each vector x in Rⁿ to a vector T(x) in Rm.
- Rⁿ refers to the domain of the transformation while Rm is the codomain.
- The collection of all images from T forms the range of T.
Identity Transformation
- The identity transformation IdRⁿ maps every vector x in Rⁿ to itself, defined as IdRⁿ(x) = x, ensuring no alteration occurs to the input vector.
Matrix Transformation
- For an m x n matrix A, the associated matrix transformation T: Rⁿ → Rm is defined by T(x) = Ax, translating input vector x in Rⁿ to the output vector Ax in Rm.
Matrix Transformation Continuation
- The domain of transformation T corresponds to Rⁿ (where n equals the number of columns of A).
- The codomain relates to Rm (where m is the number of rows of A).
- The range of transformation T corresponds specifically to the column space of A.
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Description
Explore the concepts of Basis and Dimension, along with the Rank and Basis Theorems in this quiz. Test your understanding of linear transformations and how they relate to subspaces in vector spaces. Perfect for students studying linear algebra.
