True False Linear Algebra

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

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.

det(A+B) = det(A)det(B)

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. "A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column— that is, if and only if an echelon form of the augmented matrix has no row of the form [0 ... 0 b] with b nonzero."

For any scalar c, ||cv||=c||v||

False. c=-1

The solution set of a linear system w hose augmented matrix is [a1 a2 a3 b] is the same as the solution set of Ax=b if A=[a1 a2 a3].

True

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

True

Every matrix transformation is a linear transformation.

True. We have A(ax) =aA(x) and A(x+y) = Ax + Ay so the transformation x->Ax is linear.

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

False. The zero vector. Has two entries, not independent.

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 w hen the values s1,...,sn are substituted for x1, ..., xn respectively.

False: "A solution set of the system is a set of all lists (s1,...,sn) of numbers that makes each equation a true statement when the values s1,...,sn are substituted for x1, ..., xn respectively."

A 5x6 matrix has six rows

False: "If m and n are positive integers, an m x n matrix is a rectangular array of numbers w ith m row s and n columns."

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.

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

False: x1=1 and x1=0 have the same number of row s but are not row equivalent since no elementary row operation converts the first to the second.

If the columns of the square matrix A form a basis for m then A is invertible.

True

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

True

For any square matrix A, if for some nonzero vector x, Ax 0 then the rows of A are linearly dependent.

True.

For any square matrix A, if the rows are linearly independent so are the columns.

True.

Given nonzero vectors u and v, the matrix uv^T has a null space equal to all vectors perpendicular to v.

True.

If A is invertible, then the inverse of A-1 is A itself.

True.

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

True.

The Gaussian Elimination Algorithm with Partial Pivoting has multipliers no larger than one in absolute value.

True.

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

True.

The columns of the standard matrix for a linear transformation from R^n to R^m are the images under T of the columns of the nxn identity matrix .

True.

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

True.

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

True. The vertices 0,p,x +p,x form a parallelogram so the effect of adding p to x to get x + p is a parallel line to the line from zero to p.

Two fundamental questions about a linear system involve existence and uniqueness.

True: "TWO FUNDAMENTAL QUESTIONS ABOUT A LINEAR SYSTEM 1. Is the system consistent; that is, does at least one solution exist? 2. If a solution exists, is it the only one; that is, is the solution unique?"

For any mxn matrix A, the null space of A is a subspace of R^m

False

If the problem Ax=b has any solution x, then null space must be only {0}

False.

If u and v are linearly independent, and if w is in Span{u,v}, then u,v,w are linearly dependent.

True. If w is in span{u,v} it must be a linear combination of u and v.

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

False

If three vectors in R^3 lie in the same plane in R^3 then they are linearly dependent.

False

The original Gaussian Algorithm can have multipliers no larger than one in absolute value.

False

Elementary row operations on an augmented matrix can change the solution set of the associated linear system.

False.

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

False.

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

False.

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 view ed as an augmented matrix for a linear system."

A consistent system of linear equations could have no solution.

False. A consistent system has at least one solution.

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

False. A nontrivial solution need only have at least one non-zero component.

A product of invertible matrices is invertible, and the inverse of the product is the product of their inverses in the same order

False. Reverse order.

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

Every elementary row operation is reversible

True

If the problem Ax=b has any solution x, then b must be in the column space of A

True

A homogeneous solution is always consistent.

True.

A linear transformation T: R^n->R^m always maps the origin of R^n to the origin of R^m

True.

A mapping T: R^n to R^m is one-to-one if each vector in R^n maps onto a unique vector in R^m (meaning two vectors in R^n do not map to the same vector in R^m)

True.

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

True.

For an mxn matrix A and i<=i<=m, if x in the null space of A then x is orthogonal to Ai., the ith row of A .

True.

If the columns of A are linearly dependent, then det(A)=0.

True.

If the columns of an mxn matrix A span R^m, then the equation Ax=b is consistent for each b in R^m.

True.

If the columns of an nxn matrix are linearly independent then the columns must span R^n

True.

If the problem Ax=b has a solution x, then the problem HAx=Hb has the same solution x, for any matrix H.

True.

If ||u||^2 + ||v||^2 = ||u+v||^2 then u and v are orthogonal.

True.

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

True.

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

True.

The solution set of the linear system w hose augmented matrix is [a1 a2 a3 b] is the same as the solution set of the equation x1a1 + x2a2 + x3a3 = b

True.

if A is an nxn matrix, then (A^2)^T=(A^T)^2

True.

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."

If {v1, v2, v3, v4} is a linearly independent set of vectors in R^4 then {v1, v2, v3} is also linearly independent.

True. If no linear combination of the elements of {v1, v2, v3, v4} is zero then no linear combination of the elements of {v1, v2, v3} is zero.

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

True. Since A0=b=0, the equation Ax=b is also Ax=0 and thus is homogeneous.

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

True. The equation Ax=b has a solution if and only if b is a linear combination of the columns of A .

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

True. The range of the transformation x Ax is the set of all vectors of the form Ax and that equals the set of all linear combinations of the columns of A.

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 solution or infinitely many solutions;... "

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

True: "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 homogeneous system of equations can be inconsistent.

False. Since the zero vector is always a solution, a homogeneous system of equations can never be inconsistent.

If a set in R^n is linearly dependent then the set contains more than n vectors.

False. The zero vector in R^2 is dependent but doesn't have more than 2 vectors.

Every linear transformation from R^n to R^m is a matrix transformation.

True