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.
Signup and view all the answers
Each elementary matrix is invertible.
Signup and view all the answers
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.
Signup and view all the answers
If A is invertible, then the inverse of A^-1 is A itself.
Signup and view all the answers
If A = [a b c d] and ad = bc, then A is not invertible.
Signup and view all the answers
If A can be row reduced to the identity matrix, then A must be invertible.
Signup and view all the answers
If A is invertible, then elementary row operations that reduce A to the identity In also reduce A^-1 to In.
Signup and view all the answers
If the equation Ax=0 has only the trivial solution, then A is row equivalent to the n x n identity matrix.
Signup and view all the answers
If the columns of A span R^n, then the columns are linearly independent.
Signup and view all the answers
If A is an n x n matrix, then the equation Ax=b has at least one solution for each b in R^n.
Signup and view all the answers
If the equation Ax=0 has a nontrivial solution, then A has fewer than n pivot positions.
Signup and view all the answers
If A^T is not invertible, then A is not invertible.
Signup and view all the answers
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.
Signup and view all the answers
If the columns of A are linearly independent, then the columns of A span R^n.
Signup and view all the answers
If the equation Ax = b has at least one solution for each b in R^n, then the solution is unique for each b.
Signup and view all the answers
If the linear transformation (x) -> Ax maps R^n into R^n, then A has n pivot positions.
Signup and view all the answers
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.
Signup and view all the answers
An n x n determinant is defined by determinants of (n - 1) x (n - 1) submatrices.
Signup and view all the answers
The (i, j)-cofactor of a matrix A is the matrix A(ij) obtained by deleting from A its ith row and jth column.
Signup and view all the answers
The cofactor expansion of det(A) down a column is equal to the cofactor expansion along a row.
Signup and view all the answers
The determinant of a triangular matrix is the sum of the entries on the main diagonal.
Signup and view all the answers
A row replacement operation does not affect the determinant of a matrix.
Signup and view all the answers
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.
Signup and view all the answers
If the columns of A are linearly dependent, then det(A) = 0.
Signup and view all the answers
Det(A + B) = det(A) + det(B).
Signup and view all the answers
If three row interchanges are made in succession, then the new determinant equals the old determinant.
Signup and view all the answers
The determinant of A is the product of the diagonal entries in A.
Signup and view all the answers
If det(A) is zero, then two rows or two columns are the same, or a row or a column is zero.
Signup and view all the answers
Det(A^-1) = (-1)det(A).
Signup and view all the answers
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.
Signup and view all the answers
A vector is an arrow in three-dimensional space.
Signup and view all the answers
A subset H of a vector V is a subspace of V if the zero vector is in H.
Signup and view all the answers
A subspace is also a vector space.
Signup and view all the answers
Analog signals are used in the major control systems for the space shuttle, mentioned in the introduction to the chapter.
Signup and view all the answers
A vector is any element of a vector space.
Signup and view all the answers
If u is a vector in a vector space V, then (-1)u is the same as the negative of u.
Signup and view all the answers
A vector space is also a subspace.
Signup and view all the answers
R^2 is a subspace of R^3.
Signup and view all the answers
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.
Signup and view all the answers
The null space of A is the solution set of the equation Ax = 0.
Signup and view all the answers
The null space of an m x n matrix is in R^m.
Signup and view all the answers
The column space of A is in the range of the mapping x -> Ax.
Signup and view all the answers
If the equation Ax = b is consistent, then Col(A) is in R^m.
Signup and view all the answers
The kernel of a linear transformation is a vector space.
Signup and view all the answers
Col(A) is the set of all vectors that can be written as A(x) for some x.
Signup and view all the answers
A null space is a vector space.
Signup and view all the answers
The column space of an m x n matrix is in R^m.
Signup and view all the answers
Col(A) is the set of all solutions of A(x) = b.
Signup and view all the answers
Nul(A) is the kernel of the mapping x -> Ax.
Signup and view all the answers
The range of a linear transformation is a vector space.
Signup and view all the answers
The set of all solutions of a homogeneous linear differential equation is the kernel of a linear transformation.
Signup and view all the answers
A single vector by itself is linearly dependent.
Signup and view all the answers
If H = Span {b1,..., bp}, then {b1,..., bp} is a basis for H.
Signup and view all the answers
The columns of an invertible n x n matrix form a basis for R^n.
Signup and view all the answers
A basis is a spanning set that is as large as possible.
Signup and view all the answers
In some cases, the linear dependence relations among the columns of a matrix can be affected by certain elementary row operations on the matrix.
Signup and view all the answers
A linearly independent set in a subspace H is a basis for H.
Signup and view all the answers
If a finite set S of nonzero vectors span a vector space V, then some subset of S is a basis for V.
Signup and view all the answers
A basis is a linearly independent set that is as large as possible.
Signup and view all the answers
The standard method for producing a spanning set for Nul(A), described in 4.2, sometimes fails to produce a basis for Nul(A).
Signup and view all the answers
If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col(A).
Signup and view all the answers
If x is in V and if B contains n vectors, then the B-coordinate vector of x is in R^n.
Signup and view all the answers
If P(B) is the change-of-coordinates matrix, then [X]B = P(B)X, for x in V.
Signup and view all the answers
The vector spaces P^3 and R^3 are isomorphic.
Signup and view all the answers
If B is the standard basis for R^n, then the B-coordinate vector of an x in R^n is x itself.
Signup and view all the answers
The correspondence [X]B -> x is called the coordinate mapping.
Signup and view all the answers
In some cases, a plane in R^3 can be isomorphic to R^2.
Signup and view all the answers
The number of pivot columns of a matrix equals the dimension of its column space.
Signup and view all the answers
A plane in R^3 is a two-dimensional subspace of R^3.
Signup and view all the answers
The dimension of the vector space P(4) is 4.
Signup and view all the answers
If dim(V) = n and S is a linearly independent set in V, then S is a basis for V.
Signup and view all the answers
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.
Signup and view all the answers
R^2 is a two-dimensional subspace of R^3.
Signup and view all the answers
The number of variables in the equation Ax = 0 equals the dimension of Nul(A).
Signup and view all the answers
A vector space is infinite-dimensional if it is spanned by an infinite set.
Signup and view all the answers
If dim(V) = n and if S spans V, then S is a basis of V.
Signup and view all the answers
The only three-dimensional subspace of R^3 is R^3 itself.
Signup and view all the answers
The row space of A is the same as the column space of A^T.
Signup and view all the answers
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.
Signup and view all the answers
The dimensions of the row space and the column space of A are the same, even if A is not square.
Signup and view all the answers
The sum of the dimensions of the row space and the null space of A equals the number of rows in A.
Signup and view all the answers
On a computer, row operations can change the apparent rank of a matrix.
Signup and view all the answers
If B is any echelon form of A, then the pivot columns of B form a basis for the column space of A.
Signup and view all the answers
Row operations preserve the linear dependence relations among the rows of A.
Signup and view all the answers
The dimension of the null space of A is the number of columns of A that are not pivot columns.
Signup and view all the answers
The row space of A^T is the same as the column space of A.
Signup and view all the answers
If A and B are row equivalent, then their row spaces are the same.
Signup and view all the answers
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.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your knowledge with these true or false flashcards on Linear Algebra concepts related to matrix inverses and properties. Each card presents a statement for you to evaluate, aiding in your understanding of key principles in the subject. Perfect for exam preparation and revision!
