Questions and Answers
A null space is a vector space.
True
The column space of an m×n matrix is in ℝm.
True
The column space of A, Col A, is the set of all solutions of Ax = b.
False
The null space of A, Nul A, is the kernel of the mapping x↦Ax.
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The range of a linear transformation is a vector space.
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The set of all solutions of a homogeneous linear differential equation is the kernel of a linear transformation.
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The row space of A^T is the same as the column space of A.
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The null space of A is the solution set of the equation Ax = 0.
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The null space of an mxn matrix is in ℝm.
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The column space of A is the range of the mapping x --> Ax.
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If the equation Ax = b is consistent, then Col A is ℝm.
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The kernel of a linear transformation is a vector space.
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Column A is the set of all vectors that can be written as Ax for some x.
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What is the null space?
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What is the column space of a matrix?
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What is the row space?
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What is the kernel (or null space)?
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Study Notes
Null Space
- The null space of an m×n matrix A is a vector space and a subspace of ℝn.
- Defined as the solution set of the equation Ax = 0, consisting of all x mapped to the zero vector of ℝm by the transformation x → Ax.
- Represents all solutions to homogeneous linear equations.
Column Space
- The column space of an m×n matrix A is a vector space in ℝm, specifically a subspace.
- Defined as the range of the mapping x → Ax, representing all vectors b that can be expressed as Ax for some x in ℝn.
- Not all sets of b lead to a consistent equation Ax = b; consistency must hold for specific b values, not for every b.
Row Space
- The row space of A^T corresponds directly to the column space of A, indicating a duality between rows and columns in transposition.
- Comprises all linear combinations of the row vectors of A.
Kernel
- The kernel (or null space) of a linear transformation T is defined as the set of all u in V such that T(u) = 0, representing the zero vector in W.
- Is a vector space itself, emphasizing its structural properties in linear algebra.
Homogeneous Linear Differential Equations
- The kernel of the linear transformation associated with a homogeneous linear differential equation comprises solutions f that satisfy the equation by being mapped to zero.
Summary
- Null space = solutions to Ax = 0; Column space = all possible outputs Ax; Row space = linear combinations of rows; Kernel = input mapped to zero. Each component is crucial for understanding linear transformations and their properties in vector spaces.
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Test your understanding of null spaces and column spaces in linear algebra with this set of flashcards. Explore definitions and properties related to matrix transformations and their vector spaces. Perfect for students looking to strengthen their grasp on these fundamental concepts.
