Questions and Answers
What is the kernel of a linear transformation?
The set of all vectors v in V that satisfy T(v) = 0vector.
The kernel of a linear transformation T: V -> W is a subspace of the domain V.
True
What does the kernel of the transformation T: R^n -> R^m correspond to?
The solution space of Ax = 0vector.
The range of a linear transformation T: V -> W is a subspace of W.
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What is the relationship between the column space of A and the range of T: R^n -> R^m?
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Define nullity and rank of a linear transformation.
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What is the sum of the rank and nullity for a linear transformation T: V -> W?
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A linear transformation T: V -> W is one-to-one if and only if ker(T) = {0vector}.
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A linear transformation T: V -> W is onto if the rank of T is equal to the dimension of W.
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A linear transformation is one-to-one if and only if it is onto.
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What is an isomorphism in terms of linear transformations?
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Two finite-dimensional vector spaces V and W are isomorphic if they have different dimensions.
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Study Notes
Kernel of a Linear Transformation
- The kernel of a linear transformation T: V -> W consists of all vectors v in V such that T(v) = 0vector.
- Denoted as ker(T), the kernel represents the solutions to the equation T(v) = 0.
Theorem 6.3: Kernel as Subspace
- The kernel of T is a subspace of the domain vector space V.
Corollary to Theorem 6.3
- For the linear transformation T: R^n -> R^m defined by T(x) = Ax, the kernel corresponds to the solution space of the equation Ax = 0.
Theorem 6.4: Range as Subspace
- The range of a linear transformation T: V -> W is a subspace of W.
Corollary to Theorem 6.4
- In the linear transformation T: R^n -> R^m defined by T(x) = Ax, the column space of matrix A is equivalent to the range of T.
Rank and Nullity
- The nullity of T is defined as the dimension of the kernel, noted as nullity(T).
- The rank of T is defined as the dimension of the range, noted as rank(T).
Theorem 6.5: Rank-Nullity Theorem
- For a linear transformation T: V -> W mapping from an n-dimensional vector space, the sum of the dimensions of the kernel and range equals the dimension of the domain:
rank(T) + nullity(T) = n or dim(range) + dim(kernel) = dim(domain).
Theorem 6.6: One-to-One Linear Transformations
- A linear transformation T: V -> W is one-to-one (1to1) if and only if its kernel contains only the zero vector, i.e., ker(T) = {0vector}.
Theorem 6.7: Onto Linear Transformations
- For a linear transformation T: V -> W, T is onto if the rank of T matches the dimension of W.
Theorem 6.8: One-to-One and Onto Transformations
- For linear transformations where both V and W are n-dimensional, T is one-to-one if, and only if, it is onto.
Definition of Isomorphism
- A linear transformation T: V -> W is called an isomorphism if it is both one-to-one and onto.
- If an isomorphism exists between vector spaces V and W, they are said to be isomorphic.
Theorem 6.9: Isomorphic Spaces and Dimension
- Two finite-dimensional vector spaces V and W are isomorphic if they have the same dimension.
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Description
This set of flashcards covers the kernel and range of linear transformations, focusing on key definitions and theorems related to these concepts. Ideal for students studying linear algebra and preparing for exams in this area.
