4.5-4.6 T/F

The dimension of the vector space P4 is 4.

False - Any vector in the vector space P4 is a linear combination of this set of polynomials: {1, t, t^2, t^3, t^4}, indicating P4's dimension is 5.

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

False - Only the number of FREE variables indicates the dimension of Nul A.

R2 is a two-dimensional subspace of R3.

False - R3 is the set of all vectors with two entries. It is not even a subset of R3.

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

False - Row interchanges do not.

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

False - S is not a basis because it does not always contain exactly n elements. Counterexample: Take S = {[ 1 ] , [ 0 ]}. It is a set of linearly [ 0 ] [ 1 ] [ 0 ] [ 0 ] independent vectors in R3. S does not span the vector space R3 though. For instance, [ 1 ] [ 0 ] [ 2 ] is not an element in Span S. Thus, the set S is not a basis for R3, so the given general statement is false.

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

False - The corresponding columns of A form a basis, not the pivot columns of B.

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

False - The dimension of a vector space is the number of vectors in the basis of the vector space. A finite set can generate an infinite dimension vector space. The statement is false.

If B is any echelon form of A, and if B has three nonzero rows, then the first three rows of A form a basis for Row A.

False - The nonzero rows of B form a basis but the first three rows of A may not be linearly independent (and this characteristic is required to form a basis).

If dimV = n and if S spans V, then S is a basis of V.

False - The statement does not indicate if S is linearly independent. By definition of basis: "A basis of a vector space V is a linearly independent subset of V that spans V."

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

False - The statement does not say that the plane passes through the origin, and this is a characteristic that must be present in a subspace (containing the zero vector)

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

False - The sum equals the number of columns, by the Rank Theorem. Additionally, the row space dimension = the number of pivot columns and the null space dimension = the number of free variables (non pivot columns). These two numbers sum to the number of columns.

If dimV = p, then there exists a spanning set of p + 1 vectors in V.

True - A basis for V has p elements, and adding any element (for example, the zero vector) does not change the fact that the set spans V.

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

True - In A transpose, the rows become the columns, so this statement is true.

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

True - Since the pivot columns of a matrix form a basis for Col A, the number of pivot columns is equal to the number of vectors in the basis for Col A. Meaning, the dimension of Col A is equal to the number of pivot columns of A.

If there exists a linearly independent set {V1, ..., vp} in V, then dimV >= p.

True - The basis must have at least p elements in it since a linearly independent set in a subspace H can be expanded to find a basis for V. This indicates dimV >= p.

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

True - The number of non pivot columns corresponds with the number of free variables

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

True - The rows of A transpose are formed by the columns of A.

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

True - This means we can find the row space of A's echelon form, B, to find the row space of A.

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

True - because of rounding error

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 - by Theorem 9 - b 227 - "If a vector space V has a basis B = {b1, ..., bn}, then any set in V containing more than n vectors must be linearly dependent."

The only three-dimensional subspace of R3 is R3 itself.

True - by definition of subspace - In a 3-d subspace, the basis has 3 vectors, which are all linearly independent vectors of R3---- indicating the only three-dimensional subspace of R3 is R3.

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

True - by the Rank Theorem - "If a matrix A has n columns, then rankA + dimNulA = n." - p 158 Additionally, row space dimension = number of nonzero rows in echelon form = number of pivot columns = column space dimension.

V is a nonzero finite-dimensional vector space, and the vectors listed belong to V. If there exists a set {v1, ..., vp} that spans V, then dimV <= p.

True - by the Spanning Set Theorem. The set {v1, ..., vp} will not have more than p elements in it. Indicating dimV <= p. (By the theorem,we may remove elements from a spanning set to obtain a basis, so a spanning set must contain at least as many elements as the dimension of the space.)