Questions and Answers
The null space of A is the solution set of the equation Ax = 0.
True
The null space of an m x n matrix is in R^m.
False
The column space of A is the range of the mapping x -> Ax.
True
If the equation Ax = b is consistent, then Col A is R^m.
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The kernel of a linear transformation is a vector space.
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Col A is the set of all vectors that can be written as Ax for some x.
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A null space is a vector space.
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The column space of an m x n matrix is in R^m.
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Col A is the set of all solutions of Ax = b.
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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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A single vector by itself is linearly dependent.
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If H = Span{b1,...., bp}, then {b1,..., bp} is a basis for H.
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The columns of an invertible n x n matrix form a basis for R^n.
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A basis is a spanning set that is as large as possible.
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In some cases, the linear dependence relations among the columns of a matrix can be affected by certain elementary row operations on the matrix.
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A linearly independent set in a subspace H is a basis for H.
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If a finite set S of nonzero vectors spans a vector space V, then some subset of S is a basis for V.
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A basis is a linearly independent set that is as large as possible.
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The standard method for producing a spanning set for Nul A sometimes fails to produce a basis for Nul A.
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If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A.
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Unless stated otherwise, B is a basis for a vector space V. If x is in V and if B contains n vectors, then the B-coordinate vector of x is in R^n.
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Study Notes
Null Space and Column Space
- The null space of matrix A is defined as the set of solutions for the equation Ax = 0.
- The null space of an m x n matrix is in R^n, reflecting the number of columns in the matrix.
- The column space of A represents the range of the mapping x -> Ax, consisting of all possible outputs of the transformation.
Consistency and Column Space
- If the equation Ax = b is consistent, it does not guarantee that the column space of A is R^m; this depends on whether every b in R^m has a solution.
- Col A is the set of all vectors expressible as Ax for some x, encompassing all vectors that can be produced through linear combinations of A's columns.
Kernels and Linear Transformations
- The kernel of a linear transformation is a vector space, specifically a subspace of V.
- Nul A refers to the kernel of the mapping x -> Ax, which is defined by the equation Ax = 0.
Range of Linear Transformations
- The range of a linear transformation from V to W is a subspace of W, thus classifying it as a vector space.
Homogeneous Linear Equations
- The solutions of a homogeneous linear differential equation correspond to the kernel of the related linear transformation.
Linear Dependence and Bases
- A single vector is only considered linearly dependent if it is the zero vector; otherwise, it is independent.
- If H is defined as Span{b1,...., bp}, the set {b1,..., bp} may not necessarily be a basis for H.
- The columns of an invertible n x n matrix always form a basis for R^n, supporting the Invertible Matrix Theorem.
Spanning Sets and Basis Size
- A basis is not merely a spanning set; it is a linearly independent set that cannot be enlarged without losing its independence.
- Elementary row operations do not alter the linear dependence relationships among the columns of a matrix.
Subspace Bases
- A linearly independent set within a subspace H does not automatically qualify as a basis unless it spans H.
- If a finite set S of nonzero vectors spans a vector space V, a subset of S is guaranteed to be a basis for V.
B-coordinates
- Assuming B is a basis for a vector space V, any vector x in V represented relative to B yields a coordinate vector in R^n if B contains n vectors.
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Description
Test your understanding of linear algebra concepts with this true/false quiz based on Chapter 4. Questions cover null spaces, column spaces, and the properties of matrices. Perfect for reinforcing key ideas in your studies!
