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.
True
If A and B are n x n and invertible, then A^-1*B^-1 is the inverse of AB.
False
If A = [a b c d] and ab-cd != 0, then A is invertible.
False
If A is an invertible n x n matrix, then the equation Ax = b is consistent for each b in R^n.
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Each elementary matrix is invertible.
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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.
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If A is invertible, then the inverse of A^-1 is A itself.
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If A = [a b c d] and ad = bc, then A is not invertible.
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If A can be row reduced to the identity matrix, then A must be invertible.
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If A is invertible, then elementary row operations that reduce A to the identity In also reduce A^-1 to In.
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If the equation Ax=0 has only the trivial solution, then A is row equivalent to the n x n identity matrix.
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If the columns of A span R^n, then the columns are linearly independent.
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If A is an n x n matrix, then the equation Ax=b has at least one solution for each b in R^n.
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If the equation Ax=0 has a nontrivial solution, then A has fewer than n pivot positions.
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If A^T is not invertible, then A is not invertible.
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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.
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If the columns of A are linearly independent, then the columns of A span R^n.
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If the equation Ax = b has at least one solution for each b in R^n, then the solution is unique for each b.
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If the linear transformation (x) -> Ax maps R^n into R^n, then A has n pivot positions.
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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.
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An n x n determinant is defined by determinants of (n - 1) x (n - 1) submatrices.
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The (i, j)-cofactor of a matrix A is the matrix A(ij) obtained by deleting from A its ith row and jth column.
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The cofactor expansion of det(A) down a column is equal to the cofactor expansion along a row.
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The determinant of a triangular matrix is the sum of the entries on the main diagonal.
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A row replacement operation does not affect the determinant of a matrix.
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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.
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If the columns of A are linearly dependent, then det(A) = 0.
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Det(A + B) = det(A) + det(B).
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If three row interchanges are made in succession, then the new determinant equals the old determinant.
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The determinant of A is the product of the diagonal entries in A.
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If det(A) is zero, then two rows or two columns are the same, or a row or a column is zero.
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Det(A^-1) = (-1)det(A).
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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.
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A vector is an arrow in three-dimensional space.
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A subset H of a vector V is a subspace of V if the zero vector is in H.
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A subspace is also a vector space.
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Analog signals are used in the major control systems for the space shuttle, mentioned in the introduction to the chapter.
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A vector is any element of a vector space.
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If u is a vector in a vector space V, then (-1)u is the same as the negative of u.
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A vector space is also a subspace.
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R^2 is a subspace of R^3.
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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.
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The null space of A is the solution set of the equation Ax = 0.
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The null space of an m x n matrix is in R^m.
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The column space of A is in the range of the mapping x -> Ax.
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If the equation Ax = b is consistent, then Col(A) is in 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 A(x) 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 A(x) = 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 span 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), described in 4.2, 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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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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If P(B) is the change-of-coordinates matrix, then [X]B = P(B)X, for x in V.
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The vector spaces P^3 and R^3 are isomorphic.
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If B is the standard basis for R^n, then the B-coordinate vector of an x in R^n is x itself.
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The correspondence [X]B -> x is called the coordinate mapping.
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In some cases, a plane in R^3 can be isomorphic to R^2.
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The number of pivot columns of a matrix equals the dimension of its column space.
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A plane in R^3 is a two-dimensional subspace of R^3.
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The dimension of the vector space P(4) is 4.
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If dim(V) = n and S is a linearly independent set in V, then S is a basis for V.
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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.
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R^2 is a two-dimensional subspace of R^3.
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The number of variables in the equation Ax = 0 equals the dimension of Nul(A).
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A vector space is infinite-dimensional if it is spanned by an infinite set.
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If dim(V) = n and if S spans V, then S is a basis of V.
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The only three-dimensional subspace of R^3 is R^3 itself.
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The row space of A is the same as the column space of A^T.
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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.
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The dimensions of the row space and the column space of A are the same, even if A is not square.
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The sum of the dimensions of the row space and the null space of A equals the number of rows in A.
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On a computer, row operations can change the apparent rank of a matrix.
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If B is any echelon form of A, then the pivot columns of B form a basis for the column space of A.
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Row operations preserve the linear dependence relations among the rows of A.
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The dimension of the null space of A is the number of columns of A that are not pivot columns.
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The row space of A^T is the same as the column space of A.
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If A and B are row equivalent, then their row spaces are the same.
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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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