Linear Algebra 1.1-1.5 True/False

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

FALSE - The equation gives an implicit description of the 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 trivial solution is always a solution to the equation Ax = 0.

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 in x is nonzero

The solution set of Ax = b is obtained by translating the solution set of Ax = 0.

FALSE. This only applies to a consistent system.

I If the augmented matrix [A b] has a pivot position in every row, then the equation Ax = b is inconsistent

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

In some cases a matrix may be row reduced to more than one matrix in reduced row echelon form, using different sequences of row operations.

False

The echelon form of a matrix is unique

False

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

False

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

False

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

False

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

False

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

False

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

False

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

False

Span{a1,a2} contains only the line through a1 and the origin, and the line through the a2 and the origin.

False. Span(a1,a2) contains all linear combinations of a1 and a2. In the case that these vectors do not lie on the same line, then (for example) a1+a2 does not lie on the line through a1 or a2, yet it is a linear combination.

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

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

There are exactly three vectors in Span {a1,a2,a3}.

False. There are infinitely many vectors in Span(a1,a2,a3): for instance, all multiples of a1.

The equation x = x2u + x3v, with x2 and x3 free (and neither u or v a multiple of the other), describes a plane through the origin

TRUE

A homogeneous equation is always consistent.

TRUE - The trivial solution is always a solution.

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

TRUE. If the zero vector is a solution then b = Ax = A0 = 0. So the equation is Ax = 0, thus homogeneous.

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

TRUE. We can also think of adding p as sliding the vector along p.

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

True

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

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

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

True

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

True

I The solution set of the 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 A is an m × 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 m × n matrix whose columns do not span Rm, then the equation Ax = b is inconsistent for some b in Rm.

True

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

True

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

True

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

True

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

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 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

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

True

There are exactly three vectors in the set {a1,a2,a3}.

True. Think set vs span, set just means count the terms.

The solution set of the linear system whose augmented matrix [a1 a2 a3 b] is the same as the solution set of the equation x1a1+x2a2+a3x3=b.

True: both the matrix equation and the augmented matrix translate into the same system of linear equations.

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 [a1a2a3b] and the vector equation x1a1+x2a2+x3a3=b both translate into the same system of linear equations.