M340L Exam 2

4.4 15 a

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4.4 15 b

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

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

The dimension of the vector space P4 is 4

FALSE It's 5

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

FALSE for example, row interchanges mess things up

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

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

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

FALSE it may not span

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

FALSE must be consistent for all b

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

FALSE unless A is triangular

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.

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

TRUE

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

TRUE

A subspace is also a vector space.

TRUE

A vector is any element of a vector space

TRUE

A vector space is also a subspace.

TRUE

AB + AC = A(B + C)

TRUE

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

TRUE

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

TRUE

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

TRUE

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

TRUE

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

TRUE

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

TRUE

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

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

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.

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.

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

FALSE they are not affected

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

I If A and B are 3 x 3 and B = [b1 b2 b3], 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 x 3.

A product of invertible n x 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.

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

TRUE

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

TRUE

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

TRUE

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

TRUE

The kernel of a linear transformation is a vector space

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

TRUE

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

TRUE

The range of a linear transformation is a vector space.

TRUE

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

TRUE

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

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 homogeneous 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 (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 -1^i+j

4.4 15 c

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4.4 16 a

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4.4 16 b

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4.4 16 c

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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.

FALSE The converse is true, however.

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

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

FALSE It's R^n

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

FALSE It's the corresponding columns in A

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

FALSE It's the number of free variables

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

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

FALSE S must have exactly n elements

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

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

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

FALSE Swap A and B then its true

R^2 is a subspace of R^3

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

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

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

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

FALSE The second and third parts aren't stated correctly

I 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

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

FALSE They may not be linearly independent

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

FALSE This is true for product however.

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

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

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

FALSE digital signals are used

A vector space is infinite dimensional is it is spanned by an infinite set

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

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!

A single vector is itself linearly dependent

FALSE unless it is in the zero vector

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

FALSE unless the plane is through the origin

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

FALSE Equals number of columns by rank theorem

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.

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

FALSE It is the set of all b that have solutions

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

FALSE Matrix multiplication is "row by column".

(AB)C = (AC)B

FALSE Matrix multiplication is not commutative.

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

FALSE Not a subset, as before

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

TRUE

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

TRUE

If a finite set S of nonzero vectors spans a vector space V, the some subset 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

TRUE

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

TRUE

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

TRUE

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

TRUE

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

TRUE

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

TRUE