Math 207 Exam Two

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

True

If A is a 3x2 matrix, then the transformation x -> A x cannot map R^2 onto R^3.

True

If A is an invertible nxn matrix, then the equation A x=b is consistent for each b in R^n.

True

If A=[A1 A2] and B=[B1 B2], with A1 and A2 the same sizes as B1 and B2, respectively, then A+B=[A1 +B1 A2+B2].

True

If A=[ab cd] and ad=bc, then A is not invertible.

True

If A^T is not invertible, then A is not invertible.

True

If B = {v1,...vp} is a basis for a subspace H and if x=c1v1+...+ cpVp then c1...,cp are the coordinates of x relative to the basis B.

True

If B is a basis for a subspace H, then each vector in H can be written in only one way as a linear combination of the vectors in B.

True

If T:R^2->\R^2 rotates vectors about the origin through an angle theta/phi, then T is a linear transformation.

True

If the columns of A span R^n, then the columns are linearly independent.

True

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

True

If the equation A x=0 has only the trivial solution, then A is row equivalent to the nxn identity matrix.

True

If v1, ... vp are in R^n, then Span{v1,.. vp} is the same as the column space of the matrix [v1...vp].

True

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

True

The column space of a matrix A is the set Col A of all linear combinations of the columns of A.

True

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

True

The columns of the standard matrix for a linear transformation from R^n to R^m are the images of the columns of the nxn identity matrix.

True

The definition of the matrix x vector product A x is a special case of block multiplication.

True

The dimension of Col A is the number of pivot columns of A.

True

The dimension of the column space of A is rank A.

True

The null space of a matrix A is the set Nul A of all solutions to the homogeneous equation A x=0.

True

The null space of an mxn matrix is a subspace of R^n.

True

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

True

A B+AC=A(B+C)

True

The null space of an mxn matrix A is a subspace of R^n. Equivalently, the set of all solutions to a system A x=0 of m homogeneous linear equations in n unknowns is a subspace of R^n.

True

A linear transformation is a special type of function.

True.

A basis for a subspace H of R^n is:

a linearly independent set in H that spans H.

The standard matrix of a linear transformation from R^2 to R^2 that reflects points through the horizontal axis, the vertical axis, or the origin has the form a 0 0 d. Where a and d are += 1

True

The superposition principle is a physical description of a linear transformation.

True

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

True

A linear transformation T:R^n->R^m is completely determined by its effect on the columns of the n*n identity matrix.

True.

A transformation T is linear if and only if T(c1v1 +c2v2) = c1T(v1) + c2T(v2) for all v1 and v2 in the domain of T and for all scalars c1 and c2.

True.

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

True.

If a set of p vectors spans a p-dimensional subspace H of R^n, then these vectors form a basis for H.

True.

onto

A mapping T: R^n ->R^m is said to be onto R^m if each b in R^m is the image of at least one x in R^n.

one to one

A mapping T: R^n ->R^m is said to be onto R^m if each b in R^m is the image of at most one x in R^n.

The dimension of Nul A is the number of variables in the equation A x=0.

Fakse

(A B)C=(AC)B

False

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

False

A subset H of R^n is a subspace if the zero vector is in H.

False

A subspace of R^n is any set H such that (i) the zero vector is in H, (ii) u, v, and u+v are in H, and (iii) c is a scalar and c u is in H.

False

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

False

Each line in R^n is a one-dimensional subspace of R^n.

False

Every linear transformation is a matrix transformation

False

If A and B are 2x2 with columns Subscript[a, 1], Subscript[a, 2], and Subscript[b, 1], Subscript[b, 2], respectively, then AxB=[Subscript[a, 1] Subscript[b, 1] Subscript[a, 2] Subscript[b, 2]].

False

If A and B are 3x3 and B= b1 b2 b 3 then AB=[Ab1 +Ab2 +Ab3].

False

If A and B are nxn and invertible, then A^-1 B^-1 is the inverse of A B.

False

If A is an nxn matrix, then the equation A x=b has at least one solution for each b in R^n.

False

If A=[ab cd] and ab-cd !=0, then A is invertible.

False

If T:R^n->R^m is a linear transformation and if c is in R^m, then a uniqueness question is "Is c in the range of T?"

False

The column space of a matrix A is the set of solutions of Ax=b.

False

When two linear transformations are performed one after another, the combined effect may not always be a linear transformation.

False

A product of invertible nxn matrices is invertible, and the inverse of the product is the product of their inverses in the same order.

False, it is reverse order.

A mapping T:R^n->R^m is one-to-one if each vector in R^n maps onto a unique vector in R^m.

False.

A mapping T:R^n->R^m is onto R^m if every vector x in R^n maps onto some vector in R^m.

False.

If A is a 3x2 matrix, then the transformation x -> A x cannot be one-to-one.

False.

If A is an mxn matrix, then the range of the transformation x -> A x is R^m.

False.

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

False.

Not every linear transformation from R^n to R^m is a matrix transformation.

False.

The codomain of the transformation x -> Ax is the set of all linear combinations of the columns of A.

False.

The set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of R^m.

False.

If A is a 3x5 matrix and T is a transformation defined by T(x)=Ax, then the domain of T is R^3.

False. x is in R^5

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

False. It's reversed order.

A linear transformation preserves the operations of vector addition and scalar multiplication.

True

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

True

Each elementary matrix is invertible.

True

Every matrix transformation is a linear transformation.

True

A subspace of R^n is any set H in R^n that has three properties :

a. The zero vector is in H. b. For each u and v in H, the sum u+v is in H. c. For each u in H and each scalar c, the vector c u is in H.