Linear Algebra True/False

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

False, while this is true you can also have a set that has vectors that are scalar multiples of one another and this makes the set linearly dependent. So you can have multiple reasons for a linear dependency not just one. ( having a zero vector is also a reason)

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

False, x is nonzero if at least one entry is nonzero

Two fundamental questions about a linear system involve existence and uniqueness

True, Is the system consistent(does at least one solution exist) and If a solutions exists, is the solution unique?

If the augmented matrix of a system of linear equations has a pivot in the last column, then the system is inconsistent.

True, as the equation in this row will result in 0 = 1 an obvious contradiction. It's important that I'm talking about the augmented matrix here!

If the equation Ax = b is inconsistent, then b is not in the set spanned by the columns of A.

True, b is in the set spanned by the columns of A if and only if there is some vector x so that Ax = b

If the coefficient matrix of a system of linear equations has a pivot in every row, then the system is inconsistent.

False! In fact, this means the system must be consistent as it will be impossible to have a pivot in the last column of the augmented matrix.

An inconsistent system has more than one solution

False, An inconsistent system has no solutions

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

False, If the last column is a pivot row you will have a inconsistent matrix system

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

False, Pivot positions correspond to the leading ones in reduced echelon form which is unique. Thus pivot positions are also unique.

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

False, a set can be a scalar multiple of the other making it linearly dependent not linearly independent.

If there are more variables than equations in a system of linear equations, then it must be inconsistent.

False, for example [1 0 0 0 0 1 1 0 ]would correspond to an augmented matrix of a system of linear equations with 3 variables and 2 equations. Observe that it has infinitely many solutions. Geometrically, two planes can interact in a line.

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

False, if a set is said to be linearly dependent, there has to be at least one vector in the set can be written as a linear combination but it doesn't mean all need to be able to be written as a linear combination

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

False, if the system is inconsistent then it has no solutions regardless of free variables

If u, v, and w are nonzero vectors in R3, no two of which are parallel to each other ,then {u,v,w}must be linearly independent.

False, it may be that one of the vectors is a non-trivial linear combination of the other two, such as w = u + v.

If u, v, and w are nonzero vectors in R3 and w is not in span {u,v}, then {u,v,w}must be linearly independent.

False, it may be that {u,v}is linearly dependent and therefore the larger set {u,v,w}is also linearly dependent.

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, reduced row echelon form will only have one unique solution

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

False, the homogeneous solution Ax = 0 will always have a trivial solution so regardless of A to possess linearly independent columns or not, Ax = 0 always has a trivial solution

If one row in an echelon form of an augmented matrix is [00000], then the corresponding system of linear equations must be inconsistent.

False, the resulting equation in this row is 0 = 0 which is always true and doesn't necessitate the system of linear equations be inconsistent. (However, it may still be inconsistent!)

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

False, the row reduction algorithm can be applied to any matrix

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

False, the span can be visualized as a line as well such as {[1 1], [2 2]}

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

False, the weights can be any real number

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, there are only special cases where the solution set consists only one solution, so this would only work if the solution set had one solution not if there were multiple

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, there can be a combination in which there is a consistent system with infinite solutions

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

False, two matrices are row equivalent if there is a sequence of elementary row operators to transform one matrix into the other.

If v is a solution to the matrix equation Ax = b, then Ax = v.

False, when solving Ax = b we are solving for x, not b. It should read Av = b.

The echelon form of a matrix is unique

False. The echelon form of a matrix is not​ unique, but the reduced echelon form is unique.

A 5 x 6 Matrix has six rows

False. it has 5 rows and 6 columns

A homogeneous equation Ax = 0 is always consistent.

True x = 0 is a solution as A ·0 = 0.

Every elementary row operation is reversible.

True, Basic operation for matrices

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

True, basic rule for matrices

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

True, from the definition of basic variables they correspond to the columns that have leading 1's (pivot columns)

A vector b is a linear combination of a matrix A if and only if the equation Ax = b has at least one solution

True, if Ax = b has a solution x = u, then Au = u1a1+......*unan is a linear combination of the columns of A with weights from u and it is equal to b, Au = b

Two linear systems are equivalent if they have the same solution set.

True, if they have the same solution set then the two linear systems will be equal since the two solution sets can be swapped with one another with no issue

Two vectors are linearly dependent if and only if they lie on a line through the origin.

True, if two vectors line on the same line through the origin they must be linearly dependent.

If 0 is a solution to the matrix equation Ax = b, then this equation must be homogeneous.

True, if x = 0 is a solution, then A ·0 = b ⇒ 0 = b and the equation is homogeneous.

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

True, if x = 0 is a solution, then A* (vector)0 = b then (vector) 0 = b . hence Ax = b is the homogeneous equation of Ax = 0

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

True, if x and and y are linearly dependent then z is a linear combination of the vectors of x and y and thus the set will be linearly dependent.

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

True, it is literally just another way to write out a solution set

When u and v are nonzero vectors, Span {u, v} contains the line that is parallel to through u and the origin.

True, span {u,v} contains all scalar multiples of u. Since cu + 0v is a linear combination of u and v.

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, such a solution would be a list of constants c1,c2,c3 where c1*a1 + c2*a2 + c3*a3 = b. Which would imply b can be written as a linear combination of a1, a2, and a3

The columns of an m x n matrix A span Rm, then the equation Ax = b is consistent for each b in Rm.

True, the columns of A span Rm if and only if every vector B in Rm can be represented as a linear combination of the columns of A.

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

True, the general solution describes all solutions of the system

The columns of any 4 x 5 matrix are linearly dependent

True, the number of variables is greater than the number of equations thus dependent.

Reducing a matrix to echelon form is called the forward phase of the row reduction process.

True, this is the first step to achieving RREF

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

True, z is a linear combination of x and y if x and y are linearly dependent so z must be in the span