# M340L Exam 2

FALSE
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
Each column of AB is a linear combination of the columns of B using weights from the corresponding column of A.
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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.
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.
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.
(AB)C = (AC)B
FALSE
(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.
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
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 (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.
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.
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
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.
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
The determinant of A is the product of the diagonal entries in A
FALSE
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) = (-1)detA
FALSE
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
FALSE
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
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
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
R^2 is a subspace of R^3
FALSE
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
TRUE
The null space of A is the solution set of the equation Ax = 0.
FALSE
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
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
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
A single vector is itself linearly dependent
FALSE
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
A basis is a spanning set that is as large as possible.
FALSE
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
FALSE
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!
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
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
A plane in R^3 is a two dimensional subspace of R^3
FALSE
It’s 5
The dimension of the vector space P4 is 4
FALSE
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
R^2 is a two dimensional subspace of R^3
FALSE
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
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
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.
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
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
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
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

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