True/False Chapter 1 Flashcards
Every elementary row operation is reversible
true
A 5x6 matrix has 6 rows
false : 5 rows and 6 columns
The solution set of a linear system involving variables x1,…,xn is a list of numbers (s1,…,sn) that makes each equation in the system a true statement when the values s1,…sn are substituted for x1,…,xn, respectively
false : the description given applies to a single solution. The solution set consists of all possible solutions. Only in special cases does the solution set consist of exactly one solution
Two fundamental questions about a linear system involve uniqueness and existence
true
Elementary row operations on an augmented matrix never change the solution set of the associated linear system
true
Two matrices are row equivalent if they have the same number of rows
false: the definition of row equivalent requires that there exist a sequence of row operations that transform one matrix into the other
An inconsistent system has more than one solution
false : by definition, an inconsistent system has no solution
Two linear systems are equivalent if they have the same solution set
true
In some cases, a matrix can be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations
false : each matrix is row equivalent to one and only one reduced echelon matrix
The row reduction algorithm applies only to augmented matrices for a linear system
false
A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix
true
Finding a parametric description of the solution set of a linear system is the same as solving the system
true
If one row in an echelon form of an augmented matrix is [0 0 0 5 0] , then the associated linear system is inconsistent
false : might or might not be consistent
The echelon form of a matrix is unique
false : only the reduced form is unique
The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process
false : the pivot positions are determined completely by the positions of the leading entries in the nonzero rows of any echelon form obtained from the matrix
Reducing a matrix to echelon form is called the forward phase of the row reduction process
true
Whenever a system has free variables, the solution set contains many solutions
false: the existence of at least one free solution is not related to the presence or absence of free variables
A general solution of a system is an explicit description of all solutions of the system
true
The point in the plane corresponding to (-2, 5) and (-5, 2) lie on a line through the origin
false : if (-5, 2) were on a line through (-2, 5) and the origin, then (-5, 2) would have to be a multiple of (-2, 5), which is not the case
An example of a linear combination of vectors v1 and v2 is the vector (1/2)v1
true
The solution set of the linear system whose augmented matrix is [a1 a2 a3 b] is the same as the solution set of the equation x1a1 + x2a2 + x3a3 = b
true
The set Span {u, v} is always visualized as a plane through the origin
false : the statement is often true, but Span {u, v} is not a plane when u is a multiple of v, or when u is the zero vector
Any list of five real numbers is a vector in R5
true
The vector u results when a vector u-v is added to the vector v
true
The weights c1,…cp in a linear combination c1v1 + … + cpvp cannot all be zero
false
When u and v are nonzero vectors, Span{u, v} contains the line through u and the origin
true
Asking whether the linear system corresponding to an augmented matrix [a1 a2 a3 b] has a solution amounts to asking whether b is in Span{a1, a2, a3}
true
The equation Ax=b is referred to as a vector equation
false : it is a matrix equation
A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax=b has at least one solution
true
The equation Ax=b is consistent if the augmented matrix [A b] has a pivot in every row
false
The first entry in the product Ax is a sum of products
true
If the columns of an mxn matrix A span Rm, then the equation Ax=b is consistent for each b in Rm
true
If A is an mxn matrix and if the equation Ax=b is inconsistent for some b in Rm, then A cannot have a pivot position in every row
true
Every matrix equation Ax=b corresponds to a vector equation with the same solution set
true
Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x
true
the solution set of a linear system whose augmented matrix is [a1 a2 a3 b] is the same as the solution set of Ax = b, if A = [a1 a2 a3]
true
If the equation Ax=b is inconsistent, then b is not in the set spanned by the columns of A
true : saying that b is not in the set spanned by the columns of A is the same as saying that b is not a linear combination of the columns of A
If the augmented matrix [A b] has a pivot position in every row, then the equation Ax=b is inconsistent
false
If A is an mxn matrix whose columns do not span Rm, then the equation Ax=b is inconsistent for some b in Rm
true
A homogeneous equation is always consistent
true
The equation Ax=0 gives an explicit description of its solution set
false : the equation Ax=0 gives an implicit description of its solution set
The homogeneous equation Ax=0 has the trivial solution if and only if the equation has at least one free variable
false : the equation Ax=0 always has the trivial solution
The equation x = p + tv describes a line through v parallel to p
false : the line goes through p parallel to v
The solution set of Ax=b is the set of all vectors of the form w = p + vn, where vn is any solution of the equation Ax=0
false : the solution set could be empty, the statement is true only when there exists a vector p such that Ap=b
If x is a nontrivial solution of Ax=0, then every entry of x is nonzero
false : A nontrivial solution of Ax=0 is any nonzero x that satisfies the equation
The equation x=x2u+x3v, with x2 and x3 free (and neither u nor v a multiple of the other), describes a plane through the origin
true
The equation Ax=b is homogeneous is the zero vector is a solution
true : if the zero vector is a solution, then b=Ax=A0=0
The effect of adding p to a vector is to move the vector in a direction parallel to p
true
The solution set of Ax=b is obtained by translating the solution set of Ax=0
false : the statement is true only when the solution set of Ax=0 is nonempty
The columns of a matrix A are linearly independent if the equation Ax=0 has the trivial solution
false : a homogeneous system always has the trivial solution
If S is a linearly independent set, then each vector is a linear combination of the other vectors in S
false
The columns of any 4x5 matrix are linearly dependent
true
If x and y are linearly independent, and if {x, y, z} is linearly dependent, then z is in Span{x, y}
true
Two vectors are linearly dependent if and only if they lie on a line through the origin
true
If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent
false
If x and y are linearly independent, and if z is in Span{x, y}, then {x, y, z} is linearly dependent
true
If a set in Rn is linearly dependent, then the set contains more vectors than there are entries in each vector
false
A linear transformation is a special type of function
true
If A is a 3x5 matrix and T is a transformation defined by T(x)=Ax, then the domain of T is R3
false : the domain is R5
If A is an mxn matrix, then the range of the transformation x–> Ax is Rm
false : the range in the set of all linear combinations of the columns of A
Every linear transformation is a matrix transformation
false
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
Every matrix transformation is a linear transformation
true
The codomain of the transformation x–>Ax is the set of all linear combinations of the columns of A
false : if A is an mxn matrix, the codomain is Rm
If T:Rn–>Rm is a linear transformation and if c (vector) is in Rm, then a uniqueness question is “Is c in the range of T?”
false : the question is an existence question
A linear transformation preserves the operations of vector addition and scalar multiplication
true
The superposition principle is a physical description of a linear transformation
true
A linear transformation T:Rn–>Rm is completely determined by its effect on the columns of the nxn identity matrix
true
If T:R2–>R2 rotates vectors about the origin through an angle §, then T is a linear transformation
true
When two linear transformations are performed one after another, the combined effect may not always be a linear transformation
false
A mapping T:Rn–>Rm is onto Rm if every vector x in Rn maps onto some vector in Rm
false : any function from Rn to Rm maps each vector onto another vector
If A is a 3x2 matrix, then the transformation x–>Ax cannot be one-to-one
false
Not every linear transformation from Rn to Rm is a matrix transformation
false
The columns of the standard matrix from a linear transformation from Rn to Rm are the images of the columns of the nxn identity matrix
true
The standard matrix of a linear transformation from R2 to R2 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 + or - 1
true
A mapping T : Rn–>Rm is one-to-one if each vector in Rn maps onto a unique vector in Rm
false : any function from Rn to Rm maps a vector onto a single (unique) vector
If A is a 3x2 matrix, then the transformation x–>Ax cannot map R2 onto R3
true
