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76 Cards in this Set
- Front
- Back
In order for a matrix B to be the inverse of A, both equations AB = I and BA = I must be true.
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true
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If A and B are n x n and invertible, then A^-1B^-1 is the inverse of AB.
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false, B^-1A^-1 is the inverse
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If A = [ a b
c d ] and ab - cd does not equal 0, then A is invertible. |
false, A is invertible if ad-cb does not equal 0.
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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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true
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Each elementary matrix is invertible.
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true
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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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false, in reverse order
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If A is invertible, then the inverse of A^-1 is A itself.
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true
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If A = [ a b
c d ] and ad = bc, then A is not invertible. |
true
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If A can be row reduced to the identity matrix, then A must be invertible.
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true
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If A is invertible, then elementary row operations that reduce A to the identity In(eye-subn) also reduce A^-1 to In(eye-subn).
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false, it reduces In(eye subn) to A^-1
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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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true
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If the columns of A span Rn, then the columns are linearly independent.
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true
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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 Rn.
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false, this is only true for invertible matrices.
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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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true
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If A transpose is not invertible, then A is not invertible.
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true
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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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true
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If the columns of A are linearly independent, then the columns of A span Rn.
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true
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If the equation Ax = b has at least one solution for each b in Rn, then the solution is unique for each b.
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true
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If the linear transformation x --> Ax maps Rn into Rn, then A has n pivot positions.
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false, the transformation x --> Ax maps Rn to Rn on ALL square matrices, but that doesn't mean every square matrix has n pivot positions.
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If there is a b in Rn such that the equation Ax = b is inconsistent, then the transformation x --> Ax is not one-to-one.
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true
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A row replacement operation does not affect the determinant of a matrix.
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True
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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.
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False. If we scale any rows when getting the echelon
form, we change the determinant. |
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If the columns of A are linearly dependent, then det A = 0.
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True
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det(A + B) = det A + det B
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False. detAB = (detA)(detB)
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If two row interchanged are made in succession, then the new determinant equals the old determinant.
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True
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The determinant of A is the product of the diagonal entries in A.
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False. The determinant equals the diagonal of A only when A has been reduced to triangular form.
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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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False. ad - bc could equal 0 but that doesn't mean that two rows or two columns are the same or that a row or a column is 0.
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det A transpose = (-1) det A
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False, detA transpose = detA
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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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False, the 0 vector in V is the function f whose values f(t) are 0 for all t in R.
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A vector is an arrow in 3D space
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False, not every vector is an arrow.
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A subset H of a vector space V is a subspace of V if the zero vector is in H
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False, the subset has to contain the 0 vector, have closed addition, and have closed multiplication
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A subspace is also a vector space
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True
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Analogue signals are used in the major control systems for the space shuttle, mentioned in the introduction to the chapter
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False, digital signals are used
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A vector is any element of a vector space.
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True
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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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True
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A vector space is also a subspace.
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True
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R2 is a subspace of R3
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False, R2 is not even a subspace of R3
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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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True
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The null space of A is the solution set of the equation Ax = 0
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True
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The null space of an m x n matrix is in Rm
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False, it's in Rn
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The column space of A is the range of the mapping x --> Ax
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True
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If the equation Ax = b is consistent, then Col A is Rm for an m x n matrix.
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False, it has to be consistent for EVERY b, not just b
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The kernel of a linear transformation is a vector space.
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True
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Col A is the set of all vectors that can be written as Ax for some x.
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True
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A null space is a vector space
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True
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The column space of an m x n matrix is in Rm
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True
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Nul A is the kernel of the mapping
x --> Ax |
True
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The range of a linear transformation is a vector space
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True
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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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True
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A single vector by itself is linearly dependent.
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False, the zero vector 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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False, the set {b1.....bp} also has to be linearly independent.
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The columns of an invertible n x n matrix form a basis for Rn
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True
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A basis is a spanning set that is as large as possible
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False, it is a spanning set that is as small 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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False, elementary row ops do not affect the linear dependence relations among the columns of a matrix
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A linearly independent set in a subspace H is a basis for H
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False, the set must also coincide with H
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If a finite set S of nonzero vectors spans a vector space V, then some subset of S is a basis for V
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True
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A basis is a linearly independent set that is as large as possible
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True
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The standard method for producing a spanning set for Nul A, described in Section 4.2, sometimes fails to produce a basis for Nul A
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False, it always produces a basis
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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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False, the pivot columns of A form a basis for Col A, not B
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If x is in V and if B contains n vectors, then the B-coordinate vector of x is in Rn
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True
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If PsubB is the change-of-coordinates matrix, then [x]subB = PsubBx, for x in V
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False, x = PsubB[x]subB
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The vector spaces Psub3 and R3 are isomorphic
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False, P3 is isomorphic to R4
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If B is the standard basis for Rn, then the B-coordinate vector of an x in Rn is x itself
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True
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The correspondence [x]subB --> x is called the coordinate mapping.
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False, the coordinate mapping is x --> [x]subB
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In some cases, a plane in R3 can be isomorphic to R2
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True, if the plane passes through the origin
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The number of pivot columns of a matrix equals the dimension of its column space
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True
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A plane in R3 is a 2-dimensional subspace of R3
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False, the plane must pass through the origin
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The dimension of the vector space Psub4 is 4
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False, it's 5
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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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False, the set S must also have n elements
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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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True
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R2 is a two-dimensional subspace of R3
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False, the set R2 is not even a subspace of R3
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The number of variables in the equation Ax = 0 equals the dimension of Nul A
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False, the number of free variables is equal to 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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False, a basis could still have only finitely many elements, which would make the vector space finite-dimensional
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If dim V = n and if S spans V, then S is a basis for V
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False, the set S must also have n elements
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The only three-dimensional subspace of R3 is R3 itself
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True
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Col A is the set of all solutions of
Ax = b |
False, Col A is the set of all linear combinations of the columns of A
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