True/False Exam 2

If A and B are n × n and invertible, then A−1B−1 is the inverse of AB.

FALSE AB−1 = B−1A−1.

Analogue signals are used in the major control systems for the space shuttle, mentioned in the introduction to the chapter.

FALSE Digital signals

The standard method for producing a spanning set for Nul A, described in this section, sometimes fails to produce a basis.

FALSE IT NEVER FAILS

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.

FALSE If we scale any rows when getting the echelon form, we change the determinant.

A basis is a spanning set that is as large as possible.

FALSE It is is too large, then it is no longer linearly independent.

The determinant of a triangular matrix is the sum of the entries of the main diagonal.

FALSE It is the product of the diagonal entries.

ColA is the set of all solutions of Ax=b

FALSE It is the set of all b that have solutions.

A linearly independent set in a subspace H is a basis for H.

FALSE It may not span

If A and B are 2×2 with columns a1,a2 and b1,b2 then AB = [a1b1, a2b2].

FALSE Matrix multiplication is "row by column".

If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A.

FALSE Must look at corresponding columns in A.

The null space of an m × n matrix is in Rm

FALSE Rn

Each column of AB is a linear combination of the columns of A using weights from the corresponding column of B.

FALSE Swap A and B then its true

The (i,j)-cofactor of a matrix A is the matrix Aij obtained by deleting from A its ith row and jth column.

FALSE The cofactor is the determinant of this Aij times −1i+j

If det A is zero, then two rows or two columns are the same, or a row or a column is zero.

FALSE The converse is true, however.

R2 is a subspace of R3.

FALSE The elements in R2 aren't even in R3.

AsubsetH ofavectorspaceV isasubspaceofV ifthe 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.

FALSE The second and third parts aren't stated correctly.

The transpose of a product of matrices equals the product of their tranposes in the same order.

FALSE The transpose of a product of matrices equals the product of their tranposes in the reverse order.

If A is invertible, then elementary row operations then reduce A to to the identity also reduce A−1 to the identity.

FALSE They also reduce the identity to A−1

In some cases, the linear dependence relations among the columns of a matrix can be affected by certain elementary row operations on the matrix.

FALSE They are not affected.

If H =Span{b1,...,bn} then {b1,...,bn} is a basis for H.

FALSE They may not be linearly independent

A vector is an arrow in three-dimensional space.

FALSE This is an example of a vector, but there are certainly vectors not of this form.

IfAandBare3×3andB=[b1b2b3],then AB = [Ab1 + Ab2 + Ab3].

FALSE This is right but there should not be +'s in the solution. Remember the answer should also be 3 × 3.

det(A + B)= detA + detB.

FALSE This is true for product however.

A subset H of a vector space V, is a subspace of V if the zero vector is in H

FALSE We also need the set to be closed under addition and scalar multiplication.

The cofactor expansion of det A down a column is the negative of the cofactor expansion along a row.

FALSE We can expand down any row or column and get same determinant.

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 We need f(t) = 0 for all t.

det(AT ) = (−1)detA.

FALSE det(AT ) =detA when A is n × n.

(AB)C = (AC)B

FALSE multiplication matrices are not cumulative

The determinant of A is the product of the diagonal entries in A.

FALSE unless A is triangular

IfA is an n×n matrix then the equation Ax=b has at least one solution for each b in Rn.

FALSE we need to know more about A like if it in invertible (or anything else in Thm 8)

(AB)T =AT ∗BT

FALSE(AB)T =BT ∗AT

A product of invertible n × n matrices is invertible, and the inverse of the product of their matrices in the same order.

FALSE. It is invertible, but the inverses in the product of the inverses in the reverse order.

If the linear transformation x → Ax maps Rn into Rn then A has n pivot points.

FALSE. Since A is n × n the linear transformation x → Ax maps Rn into Rn. This doesn't tell us anything about A.

A single vector is itself linearly dependent.

False unless zero vector

IfA= c d andab−cd̸=0,thenAisinvertible.

False. A is invertible is ad − bc ̸= 0

In order for a matrix B to be the inverse of A, both equations AB = I and BA = I must be true.

RUE We'll see later that for square matrices AB=I then there is some C such that BC=I

A basis is a linearly independent set that is as large as possible.

TRUE

If A is invertible, then the inverse of A−1 is A itself.

TRUE

If the columns of A are linearly independent, then the columns of A span Rn.

TRUE

If the equation Ax = b has at least solution for each b in Rn, then the solution is unique for each b.

TRUE

IfthereisabinRn suchthattheequationAx=bis inconsistent, then the transformation x → Ax is not one-to-one

TRUE

Nul A is the kernel of the mapping x → Ax.

TRUE

The column space of A is the range of the mapping x → Ax

TRUE

The column space of an m × n matrix is in Rm

TRUE

The null space is a vector space.

TRUE

The null space of A is the solution set of the equation Ax = 0.

TRUE

The second row of AB is the second row of A multiplied on the right by B.

TRUE

The set of all solutions of a homogenous linear differential equation is the kernel of a linear transformation.

TRUE

The transpose of a sum of matrices equals the sum of their transposes.

TRUE

The range of a linear transformation is a vector space.

TRUE It's a subspace(check), thus vector space.

The kernel of a linear transformation is a vector space.

TRUE To show this we show it is a subspace

If the columns of A are linearly dependent, then det A = 0.

TRUE (For example there is a row without a pivot so must be a row of all zeros.)

A vector space is also a subspace.

TRUE (Its always a subspace of itself, at the very least.)

If the columns of A span Rn, then the columns are linearly independent

TRUE Again from Thm 8. Also is n vector span Rn they must be linearly independent.

If AT is not invertible, then A is not invertible

TRUE Also from the all false part of theorem 8.

If two row interchanges are made in succession, then the new determinant equals the old determinant.

TRUE Both changes multiply the determinant by -1 and -1*-1=1.

If u is a vector in a vector space V, then (−1)u is the same as the negative of u.

TRUE By definition of "negative" u + −u = 0 But 0 = (1 + (−1)) ∗ u = 1 ∗ u + (−1)u = u + (−1)u so (−1)u must be negative u.

If the equation Ax = 0 has only the trivial solution, then A is row equivalent to the n × n identity matrix.

TRUE From Thm 8

AB+AC=A(B+C)

TRUEMatrixmultiplicationdistributes over addition.

An n × n determinant is defined by determinants of (n − 1) × (n − 1) submatrices.

TRUEish I am a little unhappy about the defined by term in here since they are not completely defined by these submatrice

If the equation Ax = b is consistent, then Col A is Rm

false must be consistent for all b

A row replacement operation does not affect the determinant of a matrix.

TRUE Just make sure you don't multiply the row you are replacing by a constant.

Col A is the set of a vectors that can be written as Ax for some x.

TRUE Remember that Ax gives a linear combination of columns of A using x entries as weights.

AT + BT = (A + B)T

TRUE See properties of transposition. Also should be able to think through to show this. When we add we add corresponding entries, these will remain corresponding entries after transposition.

If there is an n×n matriX D such that AD=I, then there is also an n×n matrix C such that CA=I.

TRUE THM 8

If A can be row reduced to the identity matrix, then A must be invertible.

TRUE The algorithm presented in this chapter tells us how to find the inverse in this case.

The columns of an invertible n × n matrix form a basis for Rn

TRUE They are linerly independent and span Rn.

If the equation Ax = 0 has a nontrivial solution, then A has fewer than n pivot positions.

TRUE This comes from the "all false" part of THM 8.

A subspace is also a vector space.

TRUE This is the definition of subspace, a subset that satisfies the vector space properties.

A vector is any element of a vector space.

TRUE This is the definition.

If a finite set S of nonzero vectors spans a vector space V , the some subset is a basis for V .

TRUE by Spanning Set Theorem

IfA= c d andad=bc,thenAisnotinvertible

TRUE. A is invertible is ad − bc ̸= 0 but if ad = nc then ad −bc = 0 so A is not invertible.

If A is an invertible n × n matrix, then the equation Ax = b is consistent for each b in Rn.

TRUE. Since A is invertible we have that x = A−1b

Each elementary matrix is invertible

True. Let K be the elementary row operation required to change the elementary matrix back into the identity. If we preform K on the identity, we get the inverse.