Linear Algebra (True/False)

T or F. A homogeneous system of equations can be inconsistent.

F. x = 0 is always a solution of a homogeneous system.

True or False. The solution set of A x = b is the set of all vectors of the form w = p + vh , where vh is any solution of the equation A x = 0.

False. Need to add: "where p is a solution of A x = b".

The span of any two nonzero vectors in R 3 is a plane.

False. Nonzero u, v in R3 such that u k v =⇒ span {u, v} = span {u} = a line.

T or F: 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. Row echelon reduction results in unique marix

True or False: row operations are not reversible.

False. Row operations are reversible.

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

False. Row reduction results in a unique matrix.

T or F. 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. The equation corresponding to this row is 5x4 = 0, which is consistent.

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

False. The rightmost column contains a pivot, means the last equation in reduced form is 0 = 1, which is inconsistent.

True or false: If a system of equations is consistent, then it cannot have any free variables.

False. You can have consistent systems with free variables.

What does linearly dependent mean in terms of possible solutions?

It means that a non-trivial solution exists.

What does linearly independent mean in terms of possible solutions?

It means that only the trivial solution exists.

T or F. If A x = b is consistent, then the solution set of A x = b is obtained by translating the solution set of A x = 0.

T. (Theorem 6)

T or F. The columns of a matrix A are linearly independent if the equation A x = 0 has the trivial solution.

T. If columns re limearly independent, then only trivial combination of columns will give 0.

T or F. The equation A x = b is homogeneous if the zero vector is a solution.

T. If x=0 is a solution, then b = Ax = 0, so it is homogeneous.

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

True

T or F. A vector b is a linear combination of the columns of a matrix A if and only if the equation A x = b has at least one solution.

True

T or F. Any list of five real numbers is a vector in R5.

True

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

True

T or F. If the equation A x = b is consistent, then b is in the set spanned by the columns of A .

True

T or F. The reduced echelon form of a matrix is unique.

True

T or F. The solution set of a linear system whose augmented matrix is [a1 a2 a3 b] is the same as the solution set of A x = b, if A = [a1 a2 a3].

True

T or F. 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 x1 a1 + x2 a2 + x3 a3 = b.

True

4 If A is an m × n matrix and if the equation Ax = b is inconsistent for some b in R m, then A cannot have a pivot position in every row.

True. Ax = b has no solution for some b in Rm =⇒ b is pivot in A b for some b in Rm =⇒ not all rows of A are pivot.

If A is an m × n matrix whose columns do not span R m, then the equation Ax = b is inconsistent for some b in R m.

True. Columns of A do not span Rm =⇒ b cannot not be a linear combination of columns of A for some b ∈ Rm =⇒ Ax = b has no solution for some b ∈ Rm.

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

True. Columns of A span Rm =⇒ b is a linear combination of columns of A for all b ∈ Rm =⇒ Ax = b has solutions for all b ∈ Rm.

T or F. If A is an m x n matrix and if the equation A x = b is inconsistent for some b in Rm , then A cannot have a pivot position in every row. T. If A has a pivot in every row, then the equation is consistent.

True. If A has a pivot in every row, then the equation is consistent.

True or false: If a system of equations has more than one solution, it has infinitely many solutions.

True. Only 3 options: none, 1, or infinite.

What is the matrix equation?

Ax = b

T or F. If x is a nontrivial solution of A x = 0, then every entry in x is nonzero.

F. An example: x+y+z=0; a nontrivial solution: (1,-1,0).

A vector is a directed line segment (an arrow).

False

True or False: The equation x = p + tv describes a line through v parallel to p.

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

A finite set that contains 0 is linearly dependent.

True

A set with exactly one vector is linearly independent if and only if that vector is not 0.

True

A set with exactly two vectors is linearly independent if and only if neither vector is a scalar multiple of the other.

True

A single linear equation with two or more unknowns must always have infinitely many solutions.

True

T or F. Every matrix equation A x = b corresponds to a vector equation with the same solution set.

True. A matrix equation may be written in terms of vectors.

True or False: A 5 x 6 matrix has six rows.

False. It has 6 columns

Elementary row ops permit one equation in a linear system to be subtracted from another.

True

T or F. An example of a linear combination of vectors v1 and v2 is the vector (1/2) v1.

True. The linear combination does not have to include both vectors.

What is the vector equation?

c1V1 +c2V2 + ........ = b

T or F. If the coefficient matrix A has a pivot position in every row, then the equation A x = b is inconsistent.

F. The opposite is true

The set of all possible solutions is called the "......."

The solution set

If a matrix is in reduced row echelon form, then it is also in row echelon form.

True

If the set of vectors is linearly independent, then kv1, kv2, kv2 is also linearly independent for every nonzero scalar k.

True

If v1, ...., vn are linearly dependent nonzero vectors, then at least one vector vk is a unique linear combination of v1, ..., v(k-1)

True

True or False: if the augmented matrices of two linear systems are row equivalent, then the two systems have the same solution set.

True

True or False: two matrices are row equivalent if there is a sequence of elementary row operations that transforms one matrix into the other.

True

x-y=3 2x-2y=k The linear system cannot have a unique solution, regardless of the value of k.

True

An equation with two variables, x and y, is in what dimensions?

2

How many solutions will a system of two equations with three variables have?

Either none or infinite (it can't be just one). That's because you have equations for two planes (3 variables give you a plane). Two planes can intersect either in a line (infinite solutions) or not at all (no solutions). But they can't intersect in a point.

T or F. The equation A x = 0 gives an explicit description of its solution set.

F. I think Ax = b form gives an implicit solution, not an explicit one. An unsolved equation does not give description of its solution.

T or F. If A is an m x n matrix whose columns do not span Rm, then the equation A x = b is consistent for every b in Rm .

F. If b is not in the span, then the equation A x = b is inconsistent.

T or F. The homogeneous equation A x = 0 has the trivial solution if and only if the equation has at least one free variable.

F. If the equation has at least one free variable, then it has a nontrivial solution.

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

F. The augmented matrix may have a pivot position in b column. Then it is inconsistent.

A set containing a single vector is linearly independent.

False

A vector is an n-tuple of real numbers.

False

Every linearly dependent set contains the zero vector.

False

Every matrix has a unique row echelon form.

False

If a linear system has more unknowns than equations, then it must have infinitely many solutions.

False

If an elementary row op is applied to a matrix that is in row echelon form, the resulting matrix will still be in row echelon form.

False

If each equation is consistent linear system is multiplied through by a constant c, then all solutions to the new system can be obtained by multiplying solutions from the original system by c.

False

If every column of a matrix in row echelon form has a leading 1 then all entries that are not leading 1's are zero.

False

If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent.

False

If the reduced row echelon form of the augmented matrix for a linear system has a row of zeros, then the system must have infinitely many solutions.

False

Multiplying a linear equation through by zero is an acceptable elementary row op.

False

The functions f1 and f2 are linearly dependent if there is a real number x so that k1f1(x) k2f2(x) = 0 for some scalars k1 and k2.

False

The ith row vector of a matrix product AB can be computed by multiplying A by the ith row vector of B.

False

The set of 2x2 matrices that contain exactly two 1's and two 0's is a linearly independent in M22.

False

There is a vector space consisting of exactly two distinct vectors.

False

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

False. (u-v) + v = u; NOT (u-v) + v = v

If v1 and v2 are in R 4 and v1 is not a scalar multiple of v2, then {v1, v2} is linearly independent.

False. Choose v1 6= 0 and v2 = 0.

If v1, v2, v3, v4 are vectors in R 4 and {v1, v2, v3} is linearly independent, then {v1, v2, v3, v4} is linearly independent.

False. Choose v4 = −v3.

T or F. The weights c1,...,cp in a linear combination c1 v1 +...+ cp vp cannot all be zero.

False. Coefficients can be all zeros. Often in linear algebra there is talk about nontrivial linear combination; it does not mean that trivial linear combinations do not exist.

T or F. Another notation for the vector column −4 3 is horizontal [-4 3].

False. Column vector and row vector are different things; in other words, these are different matrices: 2x1, and 1x2.

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

False. If u and v are parallel, then the span will be a line.

T or F. The row reduction algorithm applies only to augmented matrices for a linear system.

False. If we want to solve the system, we do augmented

T or F. The equation Ax = b is referred to as a vector equation.

False. It is a matrix equation.

T or F. The points in the plane corresponding to −2 5 and −5 2 lie on a line through the origin.

False. To be on a line through origin, their x and y components must have the same ratio: y1/x1 = y2/x2; which is not the case.

T or F. 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. Usually span contains the plain in which lie u and v; except when u and v parallel (see previous question)

The span of any vector in R 3 is a line.

False. v = 0 =⇒ span {v} = {0} contains origin only.

In R2, where you have equations with two variables, what are the possible solutions and what are they represented by geometrically?

No solution - parallel lines. One solution - two lines intersecting in a point. Infinite solutions - they are the same line.

A system of linear equations has what possible solutions?

No solution, one solution, or infinite solution. So if you can show two solutions exist, then you must have infinite ones.

T or F. The first entry in the product A x is a sum of products.

T. The first entry in the product is a11 x1 + a12 x2 + ... + a1n xn

T or F A homogeneous equation is always consistent.

T. x = 0 is always a solution of a homogeneous system.

If A is a 3×3 matrix with two pivot positions, then the equation Ax = 0 has a nontrivial solu

TRUE If A has two pivot positions, then it has a row of zeros, and hence, because A is a 3 × 3 matrix, the solution Ax = 0 has at least one free variable, hence the equation Ax = 0 has a nontrivial solu

A homogeneous linear system in n unknowns whose corresponding augmented matrix has a reduced row echelon form with r leading 1's has n-r free variables.

True

A linear system whose equations are all homogeneous must be consistent.

True

A vector is any element of a vector space.

True

All leading 1's in a matrix in row echelon form must occur in different columns.

True

If T : R n → R n is a one-to-one linear transformation, then T is also onto.

True

If a homogeneous linear system of n equations in n unknowns has a corresponding augmented matrix with a reduced row echelon form containing n leading 1's, then the linear system has only the trivial solution.

True

The general solution to Ax = b is of the form x = xp + x0, where xp is a particular solution to Ax = b and x0 is the general solution to Ax = 0.

True

The set of vectors (v,kv) is linearly dependent for every scalar k.

True

If A is a 3×3 matrix such that the system Ax = 0 has only the trivial solution, then the equation Ax = b is consistent for every b in R 3 .

True.

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

True.

If the augmented matrix of the system Ax = b has a pivot in the last column, then the system Ax = b has no solution

True. (that's because there's a row of the form 0 0 · · · 0 b , where b 6= 0)

T or F. If the columns of an m x n matrix A span Rm , then the equation A x = b is consistent for each b in Rm .

True. Columns span Rm, which includes every vector b; so the equation is consistent for every b.

T or F. Whenever a system has free variables, the solution set contains many solutions.

True. Free variables may be set to any value, resulting in many solutions.

If {v1, v2, v3} are linearly independent vectors in R n , then {v1, v2} is linearly independent as well

True. Suppose av1 + bv2 = 0. Goal: We want to show a = b = 0. Now here's a clever trick: Add 0v3 = 0 to both sides of the equation. Then we get: av1 + bv2 + 0v3 = 0 In particular, if we let c = 0, then we get: av1 +bv2 +cv3 = 0 But v1, v2, v3 are linearly independent, so a = b = c = 0. In particular a = b = 0, which we wanted to show!

T or F: 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 augmented matrix corresponds to an equation a1 x1 + a2 x2 + a3 x3 = b, so b must be a linear combination of a1, a2, a3.