True/False Chapter 4 Flashcards
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 : the zero vector in V is the function f for whose values f(t) are zero for all t in R
A vector is an arrow in three-dimension space
false : an arrow in three-dimensional space is an example of a vector, but not every arrow is a vector
A subset H of a vector space V is a subspace of V if the zero vector is in H
false
A subspace is also a vector space
true
Analog 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 is any element of a vector space
true
If u is a vector space V, then (-1)u is the same as the negative of u
true
A vector space is also a subspace
true
R2 is a subspace of R3
false
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 second and third parts of the conditions are stated incorrectly. For example, part ii) does not state that u and v represent all possible elements of H
The null space of A is the solution set of the equation Ax=0
true
The null space of an mxn matrix is in Rm
false
The column space of A is the range of the mapping x–>Ax
true
If the equation Ax=b is consistent, then Col A is Rm
false : the equation must be consistent for every b
The kernel 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
A null space is a vector space
true
The column space of an mxn matrix is in Rm
true
Col A is the set of all solutions of Ax=b
false
Nul A is the kernel of the mapping x–>Ax
true
The range of a linear transformation is a vector space
true
The set of all solutions of a homogeneous linear differential equation is the kernel of a linear transformation
true
A single vector by itself is linearly dependent
false : the zero vector by itself is linearly dependent
If H = Span{b1,…,bp}, then {b1,…,bp} is a basis for H
false : the set {b1,…,bp} must also be linearly independent
The columns of an invertible nxn matrix form a basis for Rn
true
A basis is a spanning set that is as large as possible
false
In some cases, teh linear dependence relations among the columns of a matrix can be affected by certain elementary row operations on the matrix
false
A linearly independent set in a subspace H is a basis for H
false : the subspace spanned by the set must also coincide with H
If a finite set S of nonzero vectors spans a vector space V, then some subset of S is a basis for V
true : apply the spanning set theorem to V instead of H. The space V is nonzero because the spanning set uses nonzero vectors
A basis is a linearly independent set that is as large as possible
true
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
If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A
false
If x is in V and if B contains n vectors, then the B-coordinate vector of x is in Rn
true
If Pb is the change-of-coordinates matrix, then [x]b=Pbx, for x in V
false
The vector spaces P3 and R3 are isomorphic
false : P3 is isomorphic to R4
If B is the standard basis for Rn, then the B-coordinate vector of an x in Rn is x itself
true
The correspondence [x]b–>x is called the coordinate mapping
false : by definition, the coordinate mapping goes in the opposite direction
In some cases, a plane in R3 can be isomorphic to R2
true : if the plane passes through the origin, the plane is isomorphic to R2
The number of pivot columns of a matrix equals the dimension of its column space
true
A plane in R3 is a two-dimensional subspace of R3
false : the plane must pass through the origin
The dimension of the vector space P4 is 4
false : the dimension of Pn is n+1
If dim V = n and S is a linearly independent set in V, then S is a basis for V
false : the set S must also have n elements
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
R2 is a two-dimensional subspace of R3
false : the set R2 is not even a subset of R3
The number of variables in the equation Ax=0 equals the dimension of Nul A
false : the number of free variables is equal to the dimension of Nul A
A vector space is infinite-dimensional if it is spanned by an infinite set
false : a basis could still have only finitely many elements, which would make the vector space finite-dimensional
If dim V = n and if S spans V, then S is a basis of V
false : the set S must also have n elements
The only three-dimensional subspace of R3 is R3 itself
true
The row space of A is the same as the column space of A^T
true : the rows of A are identified with the columns of A^T
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 dimensions of the row space and column space of A are the same, even if A is not square
true
The sum of the dimensions of the row space and the null space of A equals the number of rows in A
false
On a computer, row operations can change the apparent rank of a matrix
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
Row operations preserve the linear dependence relations among the rows of A
false
The dimension of the null space of A is the number of columns of A that are not pivot columns
true
The row space of A^T is the same as the column space of A
true : the rows of A^T are the columns of A
