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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 (A)
The vector -V is the additive identity of V.
The vector -V is the additive identity of V.
False (B)
To subtract two vectors in R^n, subtract their corresponding components.
To subtract two vectors in R^n, subtract their corresponding components.
True (A)
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.
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.
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.
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.
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.
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).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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'.
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'.
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].
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.
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.
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.
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.
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.
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.
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