Midterm 3 True/False

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

A vector is any element of a vector space.

True

An arrow in three-dimensional space can be considered to be a vector.

True

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

True

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

False

A vector space is also a subspace.

True

A subspace is also a vector space.

True

ℝ² is a subspace of ℝ³.

False

The polynomials of degree two or less are a subspace of the polynomials of degree three or less.

True

A subset H of a vector space V is a subspace of V if the 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 null space of A is the solution set of the equation Ax = 0.

True

A null space is a vector space.

True

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

False

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

True

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

True

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

False

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

False

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

True

The kernel of a linear transformation is a vector space.

True

The range of a linear transformation is a vector space.

True

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

True

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

True

The row space of A is the same as the column space of Aᵀ.

True

The null space of A is the same as the row space of Aᵀ.

False

A single vector by itself is linearly dependent.

False

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

False

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

False

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

True

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

True

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

True

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

False

The standard method for producing a spanning set for Nul A, described in Section 4.2, sometimes fails to produce a basis for Nul A.

False

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

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

False

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

False

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

True

If x is in V and if B contains n vectors, then the B-coordinate vector of x is in ℝⁿ.

True

If B is the standard basis for ℝⁿ, then the B-coordinate vector of an x in ℝⁿ is x itself.

True

If P_B is the change-of-coordinates matrix, then [x]_B = P_B x, for x in V.

False

The correspondence [x]_B ↦ x is called the coordinate mapping.

False

The vector spaces ℙ₃ and ℝ³ are isomorphic.

False

In some cases, a plane in ℝ³ can be isomorphic to ℝ².

True

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

True

The number of variables in the equation Ax = 0 equals the nullity A.

False

A plane in ℝ³ is a two-dimensional subspace of ℝ³.

False

The dimension of the vector space ℙ₄ is 4.

False

The dimension of the vector space of signals, S, is 10.

False

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

True

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

False

The nullity of A is the number of columns of A that are not pivot columns.

True

If a set {v₁,...,vₚ} spans a finite-dimensional vector space V and if T is a set of more than p vectors in V, then T is linearly dependent.

True

A vector space is infinite-dimensional if it is spanned by an infinite set.

False

The columns of the change-of-coordinates matrix ₍C←B₎P are B-coordinate vectors of the vectors in C.

False

The columns of ₍C←B₎P are linearly independent.

True

If V = ℝⁿ and C is the standard basis for V, then ₍C←B₎P is the same as the change-of-coordinates matrix P_B introduced in Section 4.4.

True

If V = ℝ², B = {b₁, b₂}, and C = {c₁, c₂}, then row reduction of [c₁ c₂ b₁ b₂] to [I P] produces a matrix P that satisfies [x]_B = P[x]_C for all x in V.

False