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Questions and Answers
In order for a matrix B to be the inverse of A, both equations AB = I and BA = I must be true.
In order for a matrix B to be the inverse of A, both equations AB = I and BA = I must be true.
True (A)
If A and B are n x n and invertible, then A^-1*B^-1 is the inverse of AB.
If A and B are n x n and invertible, then A^-1*B^-1 is the inverse of AB.
False (B)
If A = [a b c d] and ab-cd != 0, then A is invertible.
If A = [a b c d] and ab-cd != 0, then A is invertible.
False (B)
If A is an invertible n x n matrix, then the equation Ax = b is consistent for each b in R^n.
If A is an invertible n x n matrix, then the equation Ax = b is consistent for each b in R^n.
Each elementary matrix is invertible.
Each elementary matrix is invertible.
A product of invertible n x n matrices is invertible, and the inverse of the product is the product of their inverses in the same order.
A product of invertible n x n matrices is invertible, and the inverse of the product is the product of their inverses in the same order.
If A is invertible, then the inverse of A^-1 is A itself.
If A is invertible, then the inverse of A^-1 is A itself.
If A = [a b c d] and ad = bc, then A is not invertible.
If A = [a b c d] and ad = bc, then A is not invertible.
If A can be row reduced to the identity matrix, then A must be invertible.
If A can be row reduced to the identity matrix, then A must be invertible.
If A is invertible, then elementary row operations that reduce A to the identity In also reduce A^-1 to In.
If A is invertible, then elementary row operations that reduce A to the identity In also reduce A^-1 to In.
If the equation Ax=0 has only the trivial solution, then A is row equivalent to the n x n identity matrix.
If the equation Ax=0 has only the trivial solution, then A is row equivalent to the n x n identity matrix.
If the columns of A span R^n, then the columns are linearly independent.
If the columns of A span R^n, then the columns are linearly independent.
If A is an n x n matrix, then the equation Ax=b has at least one solution for each b in R^n.
If A is an n x n matrix, then the equation Ax=b has at least one solution for each b in R^n.
If the equation Ax=0 has a nontrivial solution, then A has fewer than n pivot positions.
If the equation Ax=0 has a nontrivial solution, then A has fewer than n pivot positions.
If A^T is not invertible, then A is not invertible.
If A^T is not invertible, then A is not invertible.
If there is an n x n matrix D such that AD=I, then there is also an n x n matrix C such that CA =I.
If there is an n x n matrix D such that AD=I, then there is also an n x n matrix C such that CA =I.
If the columns of A are linearly independent, then the columns of A span R^n.
If the columns of A are linearly independent, then the columns of A span R^n.
If the equation Ax = b has at least one solution for each b in R^n, then the solution is unique for each b.
If the equation Ax = b has at least one solution for each b in R^n, then the solution is unique for each b.
If the linear transformation (x) -> Ax maps R^n into R^n, then A has n pivot positions.
If the linear transformation (x) -> Ax maps R^n into R^n, then A has n pivot positions.
If there is a b in R^n such that the equation Ax = b is inconsistent, then the transformation x-> Ax is not one-to-one.
If there is a b in R^n such that the equation Ax = b is inconsistent, then the transformation x-> Ax is not one-to-one.
An n x n determinant is defined by determinants of (n - 1) x (n - 1) submatrices.
An n x n determinant is defined by determinants of (n - 1) x (n - 1) submatrices.
The (i, j)-cofactor of a matrix A is the matrix A(ij) obtained by deleting from A its ith row and jth column.
The (i, j)-cofactor of a matrix A is the matrix A(ij) obtained by deleting from A its ith row and jth column.
The cofactor expansion of det(A) down a column is equal to the cofactor expansion along a row.
The cofactor expansion of det(A) down a column is equal to the cofactor expansion along a row.
The determinant of a triangular matrix is the sum of the entries on the main diagonal.
The determinant of a triangular matrix is the sum of the entries on the main diagonal.
A row replacement operation does not affect the determinant of a matrix.
A row replacement operation does not affect the determinant of a matrix.
The determinant of A is the product of 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 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.
If the columns of A are linearly dependent, then det(A) = 0.
If the columns of A are linearly dependent, then det(A) = 0.
Det(A + B) = det(A) + det(B).
Det(A + B) = det(A) + det(B).
If three row interchanges are made in succession, then the new determinant equals the old determinant.
If three row interchanges are made in succession, then the new determinant equals the old determinant.
The determinant of A is the product of the diagonal entries in A.
The determinant of A is the product of the diagonal entries in A.
If det(A) is zero, then two rows or two columns are the same, or a row or a column is zero.
If det(A) is zero, then two rows or two columns are the same, or a row or a column is zero.
Det(A^-1) = (-1)det(A).
Det(A^-1) = (-1)det(A).
If f is a function in the vector space V of all real-valued functions on R and if f(t) = 0 for some t, then f is the zero vector in V.
If f is a function in the vector space V of all real-valued functions on R and if f(t) = 0 for some t, then f is the zero vector in V.
A vector is an arrow in three-dimensional space.
A vector is an arrow in three-dimensional space.
A subset H of a vector V is a subspace of V if the zero vector is in H.
A subset H of a vector V is a subspace of V if the zero vector is in H.
A subspace is also a vector space.
A subspace is also a vector space.
Analog signals are used in the major control systems for the space shuttle, mentioned in the introduction to the chapter.
Analog signals are used in the major control systems for the space shuttle, mentioned in the introduction to the chapter.
A vector is any element of a vector space.
A vector is any element of a vector space.
If u is a vector in a vector space V, then (-1)u is the same as the negative of u.
If u is a vector in a vector space V, then (-1)u is the same as the negative of u.
A vector space is also a subspace.
A vector space is also a subspace.
R^2 is a subspace of R^3.
R^2 is a subspace of R^3.
A subset H of a vector space V is a subspace of V if the following conditions are satisfied: (i) the zero vector of V is in H, (ii) u, v, and u + v are in H, and (iii) c is a scalar and cu is in H.
A subset H of a vector space V is a subspace of V if the following conditions are satisfied: (i) the zero vector of V is in H, (ii) u, v, and u + v are in H, and (iii) c is a scalar and cu is in H.
The null space of A is the solution set of the equation Ax = 0.
The null space of A is the solution set of the equation Ax = 0.
The null space of an m x n matrix is in R^m.
The null space of an m x n matrix is in R^m.
The column space of A is in the range of the mapping x -> Ax.
The column space of A is in the range of the mapping x -> Ax.
If the equation Ax = b is consistent, then Col(A) is in R^m.
If the equation Ax = b is consistent, then Col(A) is in R^m.
The kernel of a linear transformation is a vector space.
The kernel of a linear transformation is a vector space.
Col(A) is the set of all vectors that can be written as A(x) for some x.
Col(A) is the set of all vectors that can be written as A(x) for some x.
A null space is a vector space.
A null space is a vector space.
The column space of an m x n matrix is in R^m.
The column space of an m x n matrix is in R^m.
Col(A) is the set of all solutions of A(x) = b.
Col(A) is the set of all solutions of A(x) = b.
Nul(A) is the kernel of the mapping x -> Ax.
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 single vector by itself is linearly dependent.
A single vector by itself is linearly dependent.
If H = Span {b1,..., bp}, then {b1,..., bp} is a basis for H.
If H = Span {b1,..., bp}, then {b1,..., bp} is a basis for H.
The columns of an invertible n x n matrix form a basis for R^n.
The columns of an invertible n x n matrix form a basis for R^n.
A basis is a spanning set that is as large as possible.
A basis is a spanning set that is as large as possible.
In some cases, the linear dependence relations among the columns of a matrix can be affected by certain elementary row operations on the matrix.
In some cases, the linear dependence relations among the columns of a matrix can be affected by certain elementary row operations on the matrix.
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 span a vector space V, then some subset of S is a basis for V.
If a finite set S of nonzero vectors span 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), described in 4.2, sometimes fails to produce a basis for Nul(A).
The standard method for producing a spanning set for Nul(A), described in 4.2, 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 x is in V and if B contains n vectors, then the B-coordinate vector of x is in R^n.
If x is in V and if B contains n vectors, then the B-coordinate vector of x is in R^n.
If P(B) is the change-of-coordinates matrix, then [X]B = P(B)X, for x in V.
If P(B) is the change-of-coordinates matrix, then [X]B = P(B)X, for x in V.
The vector spaces P^3 and R^3 are isomorphic.
The vector spaces P^3 and R^3 are isomorphic.
If B is the standard basis for R^n, then the B-coordinate vector of an x in R^n is x itself.
If B is the standard basis for R^n, then the B-coordinate vector of an x in R^n 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 R^3 can be isomorphic to R^2.
In some cases, a plane in R^3 can be isomorphic to R^2.
The number of pivot columns of a matrix equals the dimension of its column space.
The number of pivot columns of a matrix equals the dimension of its column space.
A plane in R^3 is a two-dimensional subspace of R^3.
A plane in R^3 is a two-dimensional subspace of R^3.
The dimension of the vector space P(4) is 4.
The dimension of the vector space P(4) is 4.
If dim(V) = n and S is a linearly independent set in V, then S is a basis for V.
If dim(V) = n and S is a linearly independent set in V, then S is a basis for V.
If a set {v1,..., vp} spans a finite-dimensional vector space V and if T is a set of more than p vectors in V, then T is linearly dependent.
If a set {v1,..., vp} spans a finite-dimensional vector space V and if T is a set of more than p vectors in V, then T is linearly dependent.
R^2 is a two-dimensional subspace of R^3.
R^2 is a two-dimensional subspace of R^3.
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 R^3 is R^3 itself.
The only three-dimensional subspace of R^3 is R^3 itself.
The row space of A is the same as the column space of A^T.
The row space of A is the same as the column space of A^T.
If B is any echelon form of A, and if B has three nonzero rows, then the first three rows of A form a basis for Row A.
If B is any echelon form of A, and if B has three nonzero rows, then the first three rows of A form a basis for Row A.
The dimensions of the row space and the column space of A are the same, even if A is not square.
The dimensions of the row space and the column space of A are the same, even if A is not square.
The sum of the dimensions of the row space and the null space of A equals the number of rows in A.
The sum of the dimensions of the row space and the null space of A equals the number of rows in A.
On a computer, row operations can change the apparent rank of a matrix.
On a computer, row operations can change the apparent rank of a matrix.
If B is any echelon form of A, then the pivot columns of B form a basis for the column space of A.
If B is any echelon form of A, then the pivot columns of B form a basis for the column space of A.
Row operations preserve the linear dependence relations among the rows of A.
Row operations preserve the linear dependence relations among the rows of A.
The dimension of the null space of A is the number of columns of A that are not pivot columns.
The dimension of the null space of A is the number of columns of A that are not pivot columns.
The row space of A^T is the same as the column space of A.
The row space of A^T is the same as the column space of A.
If A and B are row equivalent, then their row spaces are the same.
If A and B are row equivalent, then their row spaces are the same.
Study Notes
Inverses and Matrix Equations
- For a matrix ( B ) to be the inverse of ( A ), both ( AB = I ) and ( BA = I ) must hold true.
- If ( A ) and ( B ) are ( n \times n ) invertible matrices, ( A^{-1}B^{-1} ) is NOT the inverse of ( AB ).
- A matrix ( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} ) is not guaranteed to be invertible if ( ab - cd \neq 0 ).
Consistency of Linear Equations
- An invertible ( n \times n ) matrix ( A ) guarantees that the equation ( Ax = b ) is consistent for every ( b \in \mathbb{R}^n ).
- If ( A ) can be row reduced to the identity matrix, it is guaranteed to be invertible.
- The equation ( Ax=0 ) having only the trivial solution means ( A ) is row equivalent to the identity matrix.
Linear Independence and Spanning
- If the columns of ( A ) can span ( \mathbb{R}^n ), they are linearly independent.
- If ( A ) has fewer than ( n ) pivot positions, then the equation ( Ax=0 ) has nontrivial solutions.
- Linear dependence among the columns of a matrix results in a determinant of zero.
Determinants
- The determinant of a triangular matrix is the product (not the sum) of its diagonal entries.
- A row replacement operation does not affect the determinant of the matrix.
- If ( \text{det}(A) = 0 ), it implies some rows or columns are identical or contain zeros.
Vector Spaces and Subspaces
- A vector is considered an element of a vector space. A vector space can be infinite-dimensional if it is spanned by an infinite set.
- A subspace requires the zero vector and closure under addition and scalar multiplication.
- The null space of ( A ) represents solutions of the homogeneous equation ( Ax = 0 ).
Kernel and Column Space
- The kernel (or null space) of a linear transformation is always a vector space.
- The column space of ( A ) comprises vectors expressible as ( A(x) ) for some ( x ).
- If a matrix ( A ) is consistent with an equation ( Ax = b ), then its column space must lie in ( \mathbb{R}^m ).
Basis and Dimensions
- The columns of an invertible ( n \times n ) matrix form a basis for ( \mathbb{R}^n ).
- A basis is a linearly independent set that spans the vector space, and if a vector space dimension ( \text{dim}(V) = n ) and ( S ) is independent, ( S ) does not necessarily form a basis.
- The number of pivot columns is equal to the dimension of the column space, and the row space dimension equals the column space dimension for any matrix.
Linear Transformations
- If a linear transformation ( x \to Ax ) maps ( \mathbb{R}^n ) to itself, ( A ) must possess ( n ) pivot positions.
- If the transformation ( x \to Ax ) has an inconsistent equation ( Ax = b ), it is not one-to-one.
Rank and Echelon Forms
- Row operations preserve the linear dependence relations and the rank does not change.
- The dimension of the null space equals the number of non-pivot columns in the associated linear transformation.
Additional Concepts
- If ( B ) is any echelon form of ( A ), the pivot columns of ( B ) provide a basis for the column space of ( A ).
- The relationship between row spaces and column spaces of matrices such as ( A ) and ( A^T ) can provide valuable insights, ensuring that their dimensions remain equal.
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