FALSE
Matrix multiplication is “row by column”.
Matrix multiplication is “row by column”.
If A and B are 2 x 2 matrices with columns a1, a2, and b1, b2, respectively, then AB = [a1b1 a2b2].
FALSE
Swap A and B then its true
Swap A and B then its true
Each column of AB is a linear combination of the columns of B using weights from the corresponding column of A.
TRUE
AB + AC = A(B + C)
TRUE
A^T + B^T = (A + B)^T
FALSE
The transpose of a product of matrices equals the product of their transposes in the reverse order.
The transpose of a product of matrices equals the product of their transposes in the reverse order.
I The transpose of a product of matrices equals the product of their transposes in the same order.
FALSE This is right but there
should not be +’s in the solution. Remember the answer
should also be 3 x 3.
should not be +’s in the solution. Remember the answer
should also be 3 x 3.
I If A and B are 3 x 3 and B = [b1 b2 b3], then AB = [Ab1 + Ab2 + Ab3].
TRUE
The second row of AB is the second row of A multiplied on the right by B.
FALSE
Matrix multiplication is not commutative.
Matrix multiplication is not commutative.
(AB)C = (AC)B
FALSE
(AB)^T = B^T A^T
(AB)^T = B^T A^T
(AB)^T = A^T B^T
TRUE
The transpose of a sum of matrices equals the sum of their transposes.
FALSE.
It is invertible, but the inverses in the product of the inverses in the reverse order.
It is invertible, but the inverses in the product of the inverses in the reverse order.
A product of invertible n x n matrices is invertible, and the inverse of the product of their matrices in the same order.
TRUE
If A is invertible, then the inverse of A^-1 is A itself.
TRUE
If A = [a b; c d] and ad = bc, then A is not invertible.
TRUE
If A can be row reduced to the identity matrix, then A must be invertible.
FALSE
They also reduce the identity to A^-1
They also reduce the identity to A^-1
I If A is invertible, then elementary row operations then reduce A to to the identity also reduce A^-1 to the identity
TRUE
An n x n determinant is de fined by determinants of (n – 1) x (n – 1) submatrices.
FALSE
The cofactor is the determinant of this Aij times -1^i+j
The cofactor is the determinant of this Aij times -1^i+j
The (i , j)-cofactor of a matrix A is the matrix Aij obtained by deleting from A its ith row and jth column.
FALSE
We can expand down any row or column and get same determinant.
We can expand down any row or column and get same determinant.
The cofactor expansion of det A down a column is the negative of the cofactor expansion along a row
FALSE
It is the product of the diagonal entries.
It is the product of the diagonal entries.
The determinant of a triangular matrix is the sum of the entries of the main diagonal.
TRUE
A row replacement operation does not a ffect the determinant of a matrix.
FALSE
If we scale any rows when getting the echelon form, we change the determinant
If we scale any rows when getting the echelon form, we change the determinant
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
TRUE
If the columns of A are linearly dependent, then det A = 0.
FALSE
This is true for product however.
This is true for product however.
det(A + B) = det A + det B
TRUE
If two row interchanges are made in succession, then the new determinant equals the old determinant
FALSE
unless A is triangular
unless A is triangular
The determinant of A is the product of the diagonal entries in A
FALSE
The converse is true, however.
The converse is true, however.
If det A is zero, then two rows or two columns are the same, or a row or a column is zero.
FALSE
det(A^T) = detA when A is n x n.
det(A^T) = detA when A is n x n.
det(A^T) = (-1)detA
FALSE
we need f(t) = 0 for all t
we need f(t) = 0 for all t
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
vector in V
FALSE
This is an example of a vector, but there are certainly vectors not of this form.
This is an example of a vector, but there are certainly vectors not of this form.
A vector is an arrow in three-dimensional space.
FALSE
We also need the set to be closed under
addition and scalar multiplication.
We also need the set to be closed under
addition and scalar multiplication.
A subset H of a vector space V, is a subspace of V if the zero vector is in H
TRUE
A subspace is also a vector space.
FALSE
digital signals are used
digital signals are used
Analogue signals are used in the major control systems for the space shuttle, mentioned in the introduction to the chapter
TRUE
A vector is any element of a vector space
TRUE
If u is a vector in a vector space V, then (-1)u is the same as the negative of u.
TRUE
A vector space is also a subspace.
FALSE
The elements in R^2 aren’t even in R^3
The elements in R^2 aren’t even in R^3
R^2 is a subspace of R^3
FALSE
The second and third parts aren’t stated correctly
The second and third parts aren’t stated correctly
A subset H of a vector space V is a subspace of V if the following conditions are satis ed: (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
H, (ii)u, v and u + v are in H, and (iii) c is a scalar and cu is in H
TRUE
The null space of A is the solution set of the equation Ax = 0.
FALSE
It’s R^n
It’s R^n
The null space of an m x n matrix is in R^m
TRUE
The column space of A is the range of the mapping x -> Ax.
FALSE
must be consistent for all b
must be consistent for all b
If the equation Ax = b is consistent, then Col A is R^m
TRUE
The kernel of a linear transformation is a vector space
TRUE
Col A is the set of a vectors that can be written as Ax for some x.
TRUE
The null space is a vector space
TRUE
The column space of an m x n matrix is in R^m
FALSE
It is the set of all b that have solutions
It is the set of all b that have solutions
Col A is the set of all solutions of Ax = b
TRUE
Nul A is the kernel of the mapping x -> Ax
TRUE
The range of a linear transformation is a vector space.
TRUE
The set of all solutions of a homogeneous linear di fferential equation is the kernel of a linear transformation.
FALSE
unless it is in the zero vector
unless it is in the zero vector
A single vector is itself linearly dependent
FALSE
They may not be linearly independent
They may not be linearly independent
If H = Span {b1,…,bn} then {b1,…,bn} is a basis for H
TRUE
The columns of an invertible n x n matrix form a basis for R^n
FALSE
it is too large, then it is no longer linearly independent
it is too large, then it is no longer linearly independent
A basis is a spanning set that is as large as possible.
FALSE
they are not affected
they are not affected
In some cases, the linear dependence relations among the columns of a matrix can be a ected by certain elementary row
operations on the matrix
operations on the matrix
FALSE
it may not span
it may not span
A linearly independent set in a subspace H is a basis for H
TRUE
If a fi nite set S of nonzero vectors spans a vector space V, the some subset is a basis for V
TRUE
A basis is a linearly independent set that is as large as possible.
FALSE
it never fails!
it never fails!
The standard method for producing a spanning set for Nul A, described in this section, sometimes fails to produce a basis
FALSE
Must look at corresponding columns in A
Must look at corresponding columns in A
If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A
…
4.4 15 a
…
4.4 15 b
…
4.4 15 c
…
4.4 16 a
…
4.4 16 b
…
4.4 16 c
TRUE
The number of pivot columns of a matrix equals the dimension of its column space
FALSE
unless the plane is through the origin
unless the plane is through the origin
A plane in R^3 is a two dimensional subspace of R^3
FALSE
It’s 5
It’s 5
The dimension of the vector space P4 is 4
FALSE
S must have exactly n elements
S must have exactly n elements
If dimV = n and S is a linearly independent set in V, then S is a basis for V
TRUE
If a set {v1…vn} spans a finite dimensional vector space V and if T is a set of more than n vectors in V, then T is linearly dependent
FALSE
Not a subset, as before
Not a subset, as before
R^2 is a two dimensional subspace of R^3
FALSE
It’s the number of free variables
It’s the number of free variables
The number of variables in the equation Ax = 0 equals the dimension of Nul A
FALSE
it must be impossible to span it by a finite set
it must be impossible to span it by a finite set
A vector space is infi nite dimensional is it is spanned by an infi nite set
FALSE
S must have exactly n elements or be noted as linearly independent
S must have exactly n elements or be noted as linearly independent
If dim V = n and if S spans V. then S is a basis for V
TRUE
The only three dimensional subspace of R^3 is R^3 itself
TRUE
The row space of A is the same as the column space of A^T
FALSE
The nonzero rows of B form a basis. The fi rst three rows of A may be linear dependent.
The nonzero rows of B form a basis. The fi rst three rows of A may be linear dependent.
If B is an echelon form of A, and if B has three nonzero rows, then the first three rows of A form a basis of Row A
TRUE
The dimensions of the row space and the column space of A are the same, even if A is not square
FALSE
Equals number of columns by rank theorem
Equals number of columns by rank theorem
The sum of the dimensions of the row space and the null space of A equals the number of rows in A
TRUE
On a computer, row operations can change the apparent rank of a matrix.
FALSE
It’s the corresponding columns in A
It’s the corresponding columns in A
If B is any echelon form of A, the the pivot columns of B form a basis for the column space of A
FALSE
for example, row interchanges mess things up
for example, row interchanges mess things up
Row operations preserve the linear dependence relations among the rows of A
TRUE
The dimension of null space of A is the number of columns of A that are not pivot columns
TRUE
The row space of A^T is the same as the column space of A
TRUE
If A and B are row equivalent, then their row spaces are the same
