Podcast
Questions and Answers
A row replacement operation does not affect the determinant of a matrix.
A row replacement operation does not affect the determinant of a matrix.
True (A)
The determinant of A is the product of the pivots in any echelon form U of A, multiplied by (-1)^r, where r is the number of row interchanges made during row reduction from A to U.
The determinant of A is the product of the pivots in any echelon form U of A, multiplied by (-1)^r, where r is the number of row interchanges made during row reduction from A to U.
False (B)
If the columns of A are linearly dependent, then det A = 0.
If the columns of A are linearly dependent, then det A = 0.
True (A)
Det(A + B) = det A + det B.
Det(A + B) = det A + det B.
The set of all polynomials of the form p(t) = at^4, where a is in the set of real numbers ℝ.
The set of all polynomials of the form p(t) = at^4, where a is in the set of real numbers ℝ.
The set of all polynomials in set of ℙn such that p(0) = 0.
The set of all polynomials in set of ℙn such that p(0) = 0.
A null space is a vector space.
A null space is a vector space.
The column space of an m×n matrix is in ℝm.
The column space of an m×n matrix is in ℝm.
The column space of A, Col(A), is the set of all solutions of Ax = b.
The column space of A, Col(A), is the set of all solutions of Ax = b.
The null space of A, Nul(A), is the kernel of the mapping x --> Ax.
The null space of A, Nul(A), is the kernel of the mapping x --> Ax.
The range of a linear transformation is a vector space.
The range of a linear transformation is a vector space.
The set of all solutions of a homogeneous linear differential equation is the kernel of a linear transformation.
The set of all solutions of a homogeneous linear differential equation is the kernel of a linear transformation.
A linearly independent set in a subspace H is a basis for H.
A linearly independent set in a subspace H is a basis for H.
If a finite set S of nonzero vectors spans a vector space V, then some subset of S is a basis for V.
If a finite set S of nonzero vectors spans a vector space V, then some subset of S is a basis for V.
A basis is a linearly independent set that is as large as possible.
A basis is a linearly independent set that is as large as possible.
The standard method for producing a spanning set for Nul A sometimes fails to produce a basis for Nul A.
The standard method for producing a spanning set for Nul A sometimes fails to produce a basis for Nul A.
If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col(A).
If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col(A).
If B is the standard basis for ℝn, then the B-coordinate vector of an x in ℝn is x itself.
If B is the standard basis for ℝn, then the B-coordinate vector of an x in ℝn is x itself.
The correspondence [x]B --> x is called the coordinate mapping.
The correspondence [x]B --> x is called the coordinate mapping.
In some cases, a plane in ℝ3 can be isomorphic to ℝ2.
In some cases, a plane in ℝ3 can be isomorphic to ℝ2.
The set ℝ2 is a two-dimensional subspace of ℝ3.
The set ℝ2 is a two-dimensional subspace of ℝ3.
The number of variables in the equation Ax = 0 equals the dimension of Nul A.
The number of variables in the equation Ax = 0 equals the dimension of Nul A.
A vector space is infinite-dimensional if it is spanned by an infinite set.
A vector space is infinite-dimensional if it is spanned by an infinite set.
If dim V = n and if S spans V, then S is a basis of V.
If dim V = n and if S spans V, then S is a basis of V.
The only three-dimensional subspace of ℝ3 is ℝ3 itself.
The only three-dimensional subspace of ℝ3 is ℝ3 itself.
If there exists a set {v1,..., vp} that spans V, then dim V ≤ p.
If there exists a set {v1,..., vp} that spans V, then dim V ≤ p.
If there exists a linearly independent set {v1,..., vp} in V, then dim V ≥ p.
If there exists a linearly independent set {v1,..., vp} in V, then dim V ≥ p.
If dim V = p, then there exists a spanning set of p + 1 vectors in V.
If dim V = p, then there exists a spanning set of p + 1 vectors in V.
Flashcards are hidden until you start studying
Study Notes
Determinants and Matrix Operations
- Row replacement operations do not change the determinant of a matrix.
- In echelon form, the determinant is the product of the pivots multiplied by ((-1)^r), where (r) is the number of row interchanges.
- Linear dependence of columns results in a determinant of zero, indicating non-invertibility.
Polynomials and Subspaces
- The set of polynomials (p(t) = at^4) (with (a \in \mathbb{R})) forms a subspace of (\mathbb{P}_4); includes zero vector and is closed under addition and scalar multiplication.
- Polynomials in (\mathbb{P}_n) where (p(0)=0) also form a subspace, adhering to the same closure properties.
Null Space and Column Space
- The null space of an (m \times n) matrix (A) is a subspace of (\mathbb{R}^n).
- The column space of an (m \times n) matrix is a subspace of (\mathbb{R}^m).
- Column space (Col(A)) is defined as the set of all vectors (b) that can be expressed as (Ax).
Linear Transformations and Kernels
- The kernel of a linear transformation represents the null space.
- The range of a linear transformation is a subspace of its codomain.
Basis and Dimension
- A linearly independent set in a subspace is not necessarily a basis unless it spans the subspace.
- Any finite spanning set (S) guarantees that a subset forms a basis, confirming the Spanning Set Theorem.
- A basis is defined as a linearly independent set that is maximal.
- The dimension of the null space is defined by the number of free variables, not the total number of variables in the equation.
Relationships and Isomorphisms
- A plane in (\mathbb{R}^3) passing through the origin can be considered isomorphic to (\mathbb{R}^2).
- (\mathbb{R}^2) is not a subspace of (\mathbb{R}^3), hence not counted as such.
General Statements on Dimensions
- If a vector space (V) has a dimension (n), a spanning set must also contain exactly (n) elements to qualify as a basis.
- If dimension (dim V = p), it is possible to create a spanning set consisting of (p + 1) vectors by adding the zero vector to any existing spanning set.
Misconceptions and Clarifications
- The standard basis in (\mathbb{R}^n) correlates directly with coordinate vectors in the space.
- Equating the null space dimension with the total count of variables leads to incorrect conclusions; focus on free variables as key indicators.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
