question

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

answer

FALSE Matrix multiplication is "row by column".

question

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

answer

FALSE Swap A and B then its true

question

AB + AC = A(B + C)

answer

TRUE

question

A^T + B^T = (A + B)^T

answer

TRUE

question

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

answer

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

question

I If A and B are 3 x 3 and B = [b1 b2 b3], then AB = [Ab1 + Ab2 + Ab3].

answer

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

question

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

answer

TRUE

question

(AB)C = (AC)B

answer

FALSE Matrix multiplication is not commutative.

question

(AB)^T = A^T B^T

answer

FALSE (AB)^T = B^T A^T

question

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

answer

TRUE

question

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

answer

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

question

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

answer

TRUE

question

If A = [a b; c d] and ad = bc, then A is not invertible.

answer

TRUE

question

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

answer

TRUE

question

I If A is invertible, then elementary row operations then reduce A to to the identity also reduce A^-1 to the identity

answer

FALSE They also reduce the identity to A^-1

question

An n x n determinant is defined by determinants of (n - 1) x (n - 1) submatrices.

answer

TRUE

question

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

answer

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

question

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

answer

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

question

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

answer

FALSE It is the product of the diagonal entries.

question

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

answer

TRUE

question

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

answer

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

question

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

answer

TRUE

question

det(A + B) = det A + det B

answer

FALSE This is true for product however.

question

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

answer

TRUE

question

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

answer

FALSE unless A is triangular

question

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

answer

FALSE The converse is true, however.

question

det(A^T) = (-1)detA

answer

FALSE det(A^T) = detA when A is n x n.

question

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

answer

FALSE we need f(t) = 0 for all t

question

A vector is an arrow in three-dimensional space.

answer

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

question

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

answer

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

question

A subspace is also a vector space.

answer

TRUE

question

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

answer

FALSE digital signals are used

question

A vector is any element of a vector space

answer

TRUE

question

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

answer

TRUE

question

A vector space is also a subspace.

answer

TRUE

question

R^2 is a subspace of R^3

answer

FALSE The elements in R^2 aren't even in R^3

question

A subset H of a vector space V is a subspace of V if the following conditions are satised: (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

answer

FALSE The second and third parts aren't stated correctly

question

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

answer

TRUE

question

The null space of an m x n matrix is in R^m

answer

FALSE It's R^n

question

The column space of A is the range of the mapping x -> Ax.

answer

TRUE

question

If the equation Ax = b is consistent, then Col A is R^m

answer

FALSE must be consistent for all b

question

The kernel of a linear transformation is a vector space

answer

TRUE

question

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

answer

TRUE

question

The null space is a vector space

answer

TRUE

question

The column space of an m x n matrix is in R^m

answer

TRUE

question

Col A is the set of all solutions of Ax = b

answer

FALSE It is the set of all b that have solutions

question

Nul A is the kernel of the mapping x -> Ax

answer

TRUE

question

The range of a linear transformation is a vector space.

answer

TRUE

question

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

answer

TRUE

question

A single vector is itself linearly dependent

answer

FALSE unless it is in the zero vector

question

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

answer

FALSE They may not be linearly independent

question

The columns of an invertible n x n matrix form a basis for R^n

answer

TRUE

question

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

answer

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

question

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

answer

FALSE they are not affected

question

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

answer

FALSE it may not span

question

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

answer

TRUE

question

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

answer

TRUE

question

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

answer

FALSE it never fails!

question

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

answer

FALSE Must look at corresponding columns in A

question

4.4 15 a

answer

...

question

4.4 15 b

answer

...

question

4.4 15 c

answer

...

question

4.4 16 a

answer

...

question

4.4 16 b

answer

...

question

4.4 16 c

answer

...

question

The number of pivot columns of a matrix equals the dimension of its column space

answer

TRUE

question

A plane in R^3 is a two dimensional subspace of R^3

answer

FALSE unless the plane is through the origin

question

The dimension of the vector space P4 is 4

answer

FALSE It's 5

question

If dimV = n and S is a linearly independent set in V, then S is a basis for V

answer

FALSE S must have exactly n elements

question

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

answer

TRUE

question

R^2 is a two dimensional subspace of R^3

answer

FALSE Not a subset, as before

question

The number of variables in the equation Ax = 0 equals the dimension of Nul A

answer

FALSE It's the number of free variables

question

A vector space is infinite dimensional is it is spanned by an infinite set

answer

FALSE it must be impossible to span it by a finite set

question

If dim V = n and if S spans V. then S is a basis for V

answer

FALSE S must have exactly n elements or be noted as linearly independent

question

The only three dimensional subspace of R^3 is R^3 itself

answer

TRUE

question

The row space of A is the same as the column space of A^T

answer

TRUE

question

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

answer

FALSE The nonzero rows of B form a basis. The first three rows of A may be linear dependent.

question

The dimensions of the row space and the column space of A are the same, even if A is not square

answer

TRUE

question

The sum of the dimensions of the row space and the null space of A equals the number of rows in A

answer

FALSE Equals number of columns by rank theorem

question

On a computer, row operations can change the apparent rank of a matrix.

answer

TRUE

question

If B is any echelon form of A, the the pivot columns of B form a basis for the column space of A

answer

FALSE It's the corresponding columns in A

question

Row operations preserve the linear dependence relations among the rows of A

answer

FALSE for example, row interchanges mess things up

question

The dimension of null space of A is the number of columns of A that are not pivot columns

answer

TRUE

question

The row space of A^T is the same as the column space of A

answer

TRUE

question

If A and B are row equivalent, then their row spaces are the same

answer

TRUE

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