Linear Algebra True/ False Ch 1

A homogenous system of equations can be inconsistent

False

A mapping T: Rn --> Rm is onto Rm if every vector x in Rn maps onto some vector in Rm

False

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

False

If A is an m x n matrix, then the range of the transformation x-->Ax is Rm

False

If A is in an mxn matrix whose columns do not span Rm, then the equation Ax=b is consistent for every b in Rm

False

If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent.

False

If a set in Rn is linearly dependent, then the set contains more vectors than there are entries in each vector.

False

If every column of an augmented matrix contains a pivot, then the corresponding system is consistent.

False

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

If the coefficient matrix A has a pivot position in every row, then the equation Ax=b is inconsistent

False

The equation Ax = b is consistent if the augmented matrix [A b] has a pivot position in every row.

False

The points in the plane corresponding to vectors [-2, 5] and [-5,2] lie on a line through the origin

False

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

False

When u and v are nonzero vectors, Span {u, v} contains only the line through u and the origin, and the line through v and the origin

False

The equation Ax=b is referred to as a vector equation

False, the equation Ax=b is referred to as a matrix equation because A is a matrix.

Every linear transformation is a matrix transformation

False, though the reverse is true

In some cases, a matrix may 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. "The algorithm applies to any matrix, whether or not the matrix is viewed as an augmented matrix for a linear system."

The equation x = p + tv describes a line through v parallel to p

False. The line goes through p and is parallel to v.

Two equivalent linear systems can have different solution sets.

False: "Suppose a system is changed to a new one via row operations. By considering each type of row operation, you can see that any solution of the original system remains a solution of the new system.

A 5 x 6 matrix has 6 rows

False; 5 rows and 6 columns

Whenever a solution has free variables, the solution set contains many solutions

False; It is possible to have a free variable but the system is inconsistent, so the existence of free variables does not guarantee any solution.

The homogeneous equation Ax=0 has the trivial if and only if the equation has at least one free variable

False; The homogeneous equation Ax=0 has the NONtrivial if and only if the equation has at least one free variable

The solution set of Ax = b is the set of all vectors of the form w = p + vh, where vh is any solution of the equation Ax = 0

False; This is only true when there exists some vector p such that Ap = b.

If x is a nontrivial solution of Ax=0, then every entry in x is nonzero.

False; at least one entry is nonzero

The set Span {u, v} is always visualized as a plane through the origin

False; if u and v are multiples of one another, the span is a line

The equation Ax = 0 gives an explicit description of its solution set

False; it gives an implicit description

If S is a linearly dependent set, then each vector is a linear combination of the other vectors in S

False; only two of the vectors have to be linear combinations of one another

The vector v results when a vector u-v is added to the vector v.

False; results in u

If A is a 3 x 5 matrix and T is a transformation defined by T(x) = Ax, then the domain of T is R3

False; the domain is R5

Another notation for the vector <−4 3> as a column is (−4 3 ) across.

False; the first is a vector in R 2 , and so is a 2-by-1 matrix, while the second is a 1-by-2 matrix, and in particular is not a vector in R 2 .

The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process.

False; the pivot positions are the leading non-zero entries in the rows and have no bearing whatsoever on which row operations are used.

The columns of a matrix A are linearly independent if the equation Ax=0 has the trivial solution

False; the trivial solution is always a solution

Two matrices are row equivalent if they have the same number of rows.

False; they are row equivalent if you can get from one to the other using elementary row operations.

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; this describes one element of the solution set, not the entire set

If T : Rn --> Rm is a linear transformation and if c is in Rm, then a uniqueness question is "Is c in the range of T?"

False; this is an existence question

The weights c1, . . . , cp in a linear combination c1v1 + · · · + cpvp cannot all be zero.

False; would result in vector of 0

A general solution of a system is an explicit description of all solutions of the system.

True

A linear transformation T: Rn -> Rm always maps the origin of Rn to the origin of Rm

True

A linear transformation T: Rn -> Rm is completely determined by its effect on the columns of the n x n identity matrix

True

A linear transformation is a special type of function

True

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

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

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

An example of a linear combination of vectors v1 and v2 is the vector 1/2 v1

True

Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x.

True

Any list of five real numbers is a vector in R5

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

Elementary row operations on an augmented matrix never change the solution set of the associated linear system.

True

Every elementary row is reversible

True

Every matrix equation Ax=b corresponds to a vector equation with the same solution set.

True

Every matrix transformation is a linear transformation.

True

If Ax=b is consistent, then the solution set of Ax=b is obtained by translating the solution set of Ax=0

True

If T: R2-->R2 rotates vectors about the origin through an angle z, then T is a linear transformation

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 the equation Ax=b is consistent, then b is in the set spanned by the columns of A.

True

If three vectors in R3 lie in the same plane in R3, then they are linearly dependent.

True

The effect of adding p to a vector is to move the vector in a direction parallel to p

True

The equation Ax=b is homogeneous if the zero vector is a solution.

True

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

True

The reduced echelon form of a matrix is unique

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

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

Two fundamental questions about a linear system involve existence and uniqueness

True

if A is a 4 x 3 matrix, then the transformation x -> Ax maps R3 onto R4

True

If A is an m x n 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, if A is an mxn matrix and if the equation Ax=b is inconsistent for some b in R^m, then the equation Ax=b has no solution for some b in R^m

The first entry in the product Ax is the sum of products

True, the first entry in Ax is the sum of products of corresponding entries in x and the first entry in each column of A.

Finding a parametric description of the solution set of a linear system is the same as solving the system.

True. "Solving a system amounts to finding a parametric description of the solution set or determining that the solution set is empty."

A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix.

True. "The variables ... corresponding to pivot columns in the matrix are called basic variables."

A consistent system of linear equations has one or more solutions

True: "A system of linear equations is said to be consistent if it has either one solutionor infinitely many solutions;... "

If u and v are linearly independent, and if x is in their span, then {u, v, w} is linearly dependent

True; If z is in the Span{x, y} then z is a linear combination of the other two, which can be rearranged to show linear dependence.

The columns of any 4x5 matrix are linearly dependent

True; Theorem 8 states that any matrix with more columns than rows is linearly deendent

A homogeneous equation is always consistent

True; since the zero vector is always a solution, a homogeneous system of equations can never be inconsistent.

If x and y are linearly independent, and if {x, y,z } is linearly dependent, then z is in Span {x, y}

True; since x and y are linearly independent, and {x, y, z} is linearly dependent, it must be that z can be written as a linear combination of the other two, thus is in their span