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Questions and Answers
Two vectors in R^n are equal if and only if their corresponding components are equal.
Two vectors in R^n are equal if and only if their corresponding components are equal.
True
The vector -V is the additive identity of V.
The vector -V is the additive identity of V.
False
To subtract two vectors in R^n, subtract their corresponding components.
To subtract two vectors in R^n, subtract their corresponding components.
True
The zero vector 0 in R^n is the additive inverse of a vector.
The zero vector 0 in R^n is the additive inverse of a vector.
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To show that a set is not a vector space, it is sufficient to show that just one axiom is not satisfied.
To show that a set is not a vector space, it is sufficient to show that just one axiom is not satisfied.
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The set of all first-degree polynomials with the standard operations is a vector space.
The set of all first-degree polynomials with the standard operations is a vector space.
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The set of all pairs of real numbers of the form (0,y) with the standard operations on R^2, is a vector space.
The set of all pairs of real numbers of the form (0,y) with the standard operations on R^2, is a vector space.
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Every vector space V contains two proper subspace that are the zero subspace and itself.
Every vector space V contains two proper subspace that are the zero subspace and itself.
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If W is a subspace of R^2, then W must contain the vector (0,0).
If W is a subspace of R^2, then W must contain the vector (0,0).
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If W is a subspace of a vector space V, then it has closure under addition as defined in V.
If W is a subspace of a vector space V, then it has closure under addition as defined in V.
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If W is the subspace of a vector space V, then W is also a vector space.
If W is the subspace of a vector space V, then W is also a vector space.
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A set S ={V1, V2,..., Vk}, k>2 is linearly dependent if and only if at least one of the vectors Vi can be written as a linear combination of the other vectors in S.
A set S ={V1, V2,..., Vk}, k>2 is linearly dependent if and only if at least one of the vectors Vi can be written as a linear combination of the other vectors in S.
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If the subset S spans a vector space V, then every vector in V can be written as a linear combination of the vectors in S.
If the subset S spans a vector space V, then every vector in V can be written as a linear combination of the vectors in S.
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If dim(V) = n, then any set of n+1 vectors in V must be linearly dependent.
If dim(V) = n, then any set of n+1 vectors in V must be linearly dependent.
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If dim(V) = n, then any set of n-1 vectors in V must be linearly independent.
If dim(V) = n, then any set of n-1 vectors in V must be linearly independent.
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If dim(V) = n, then there exists a set of n-1 vectors in V that spans V.
If dim(V) = n, then there exists a set of n-1 vectors in V that spans V.
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If dim(V) = n, then there exists a set of n+1 vectors in V that span V.
If dim(V) = n, then there exists a set of n+1 vectors in V that span V.
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If an m x n matrix A is row-equivalent to an m x n matrix B, then the row space of A is equivalent to the row space of B.
If an m x n matrix A is row-equivalent to an m x n matrix B, then the row space of A is equivalent to the row space of B.
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If A is an m x n matrix of rank r, then the dimension of the solution space of Ax=0 is m-r.
If A is an m x n matrix of rank r, then the dimension of the solution space of Ax=0 is m-r.
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If an m x n matrix B can be obtained from elementary row operations on an m x n matrix A, then the column space of B is equal to the column space of A.
If an m x n matrix B can be obtained from elementary row operations on an m x n matrix A, then the column space of B is equal to the column space of A.
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The system of linear equations Ax = b is inconsistent if and only if b is the column space of A.
The system of linear equations Ax = b is inconsistent if and only if b is the column space of A.
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The column space of a matrix A is equal to the row space of A^T.
The column space of a matrix A is equal to the row space of A^T.
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The row space of a matrix A is equal to the column space of A^T.
The row space of a matrix A is equal to the column space of A^T.
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If P is the transition matrix from the basis B' to B, then P^-1 is the transition matrix from B to B'.
If P is the transition matrix from the basis B' to B, then P^-1 is the transition matrix from B to B'.
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To perform the change of basis from nonstandard basis B' to the standard basis B, the transition matrix P^-1 is simply B'.
To perform the change of basis from nonstandard basis B' to the standard basis B, the transition matrix P^-1 is simply B'.
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The coordinate matrix of p= -3 + x + 5x^2 relative to the standard basis for P2 is [p]s = [5 1 3].
The coordinate matrix of p= -3 + x + 5x^2 relative to the standard basis for P2 is [p]s = [5 1 3].
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If v is a nonzero vector in R^n, then the unit vector in the direction of v is u= llvll/v.
If v is a nonzero vector in R^n, then the unit vector in the direction of v is u= llvll/v.
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If u x v < 0, then the angle theta between u and v is acute.
If u x v < 0, then the angle theta between u and v is acute.
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The norm of the vector u is the angle between u and the positive x-axis.
The norm of the vector u is the angle between u and the positive x-axis.
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The angle theta between a vector v and the projection of u onto v is obtuse when the scalar a < 0 and acute when a > 0 where av = projvU.
The angle theta between a vector v and the projection of u onto v is obtuse when the scalar a < 0 and acute when a > 0 where av = projvU.
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A set S of vectors in an inner product space V is orthogonal when every pair of vectors in S is orthogonal.
A set S of vectors in an inner product space V is orthogonal when every pair of vectors in S is orthogonal.
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An orthonormal basis derived by the Gram-Schmidt orthonormalization process does not depend on the order of the vectors in the basis.
An orthonormal basis derived by the Gram-Schmidt orthonormalization process does not depend on the order of the vectors in the basis.
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Study Notes
Vector Equality and Operations
- Two vectors in R^n are equal if their corresponding components are equal.
- The vector -V is not the additive identity; the additive identity is the zero vector.
- To subtract vectors in R^n, directly subtract their components.
- The zero vector is not the additive inverse of a vector; it serves as the identity element instead.
Vector Spaces and Subspaces
- A set is not a vector space if any single axiom is violated.
- The set of first-degree polynomials is not classified as a vector space under standard operations.
- Pairs of real numbers of the form (0,y) form a vector space with the standard operations in R^2.
- Every vector space contains at least two proper subspaces: the zero subspace and the vector space itself.
- Subspaces must include the zero vector (0,0) and adhere to closure under addition.
Linear Dependence and Dimensionality
- A set of vectors is linearly dependent if at least one vector can be expressed as a linear combination of the others.
- If a subset spans a vector space V, every vector in V can be formed as a linear combination of that subset.
- For a vector space V with dimension n, any set of n+1 vectors must be linearly dependent.
- Not every set of n-1 vectors in a dimension n space is guaranteed to be linearly independent; this depends on the specific vectors.
Matrices and Row/Column Spaces
- If two matrices are row-equivalent, their row spaces are equivalent.
- The dimension of the solution space for the equation Ax=0 equals the number of columns minus the rank, not m-r.
- Elementary row operations do not guarantee that the column spaces of two matrices remain equal.
- A system of linear equations Ax = b is inconsistent if b is not in the column space of A.
Transitions and Changes of Basis
- The transition matrix P from basis B' to B has an inverse P^-1, which transitions from B to B'.
- The transition matrix for a change of basis to the standard basis is not simply equal to B'.
Coordinate Matrices and Norms
- The coordinate matrix relaying a polynomial to the standard basis for P2 must be checked for accuracy; placeholders represent the coefficients in a specific order.
- The unit vector in the direction of v is determined by dividing v by its norm, not by the expression wrongly stated.
- The angle between a vector and its projection can indicate whether it is acute or obtuse based on the sign of the resulting scalar that represents the projection.
Orthogonality and Orthogonal Bases
- A set of vectors is orthogonal if each vector in the set is perpendicular to every other vector in the set.
- The Gram-Schmidt process may yield a basis, but the outcome can depend on the ordering of the original vectors.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your understanding of key concepts in Linear Algebra with this True/False flashcard exam for Unit 2. Each statement challenges your knowledge of vector properties and operations in R^n. Perfect for reinforcing what you've learned in your Linear Algebra class.
