Exam 1 True/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

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

The solution set of Ax = b is the set of all vectors of the form w=p+vh wherevh isanysolutionoftheequationAx=0

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

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

FALSE it can apply to coefficient Matrices

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

FALSE it does not matter on the row interchanges, it only matters on row reduced echelon form

IfT :Rn →Rm isalineartransformationandifcisinRm, then a uniqueness question is "Is c is the range of T."

FALSE this is an existence question

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 this is only true for the trivial solution

The echelon form of a matrix is unique

FALSE unique means it only has 1 solution

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

FALSE- For example, [1, 1] , [2, 2] and [5, 4] are linearly dependent but the last is not a linear combination of the first two.

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.

Every linear transformation is a matrix transformation.

FALSE. The converse (every matrix transformation is a linear transformation) is true, however. We (probably) will see examples of when the original statement is false later.

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

FALSE. The trivial solution is always a solution.

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

FALSE. This only applies to a consistent system.

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

False. For example, in R3 [1, 2, 3] and [3, 6, 9] are linearly dependent.

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

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

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

TRUE because they can be changed back to original matrix

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

TRUE definition part of echelon form

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

TRUE it is in the same place as the pivot position is

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

TRUE the solution set does contain many solutions because the free variable can be anything

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

TRUE you need to solve the system in order to find parametric description of solution set

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 homogenous.

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

TRUE. If they lie on a line through the origin then the origin, the zero vector, is in their span thus they are linearly dependent.

The columns of any 4 × 5 matrix are linearly dependent.

TRUE. There are five columns each with four entries, thus by Thm 8 they are linearly dependent.

If T : R2 → R2 rotates vectors about the origin through an angle φ, then T is a linear transformation.

TRUE. To show this we would show the properties of linear transformations are preserved under rotations.

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 5×6 matrix has six rows

FALSE 5 rows and 6 columns

Not every linear transformation from Rn to Rm is a matrix transformation.

FALSE For a linear transformation from Rn to Rmwe se where the basis vector in Rn get mapped to. These form the standard matrix.

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

TRUE it tells the solutions

Two fundamental questions about a linear system involve existence and uniqueness

True pg. 7

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.

A mapping T :Rn →Rm is onto Rm if every vector x in Rn maps onto some vector in Rm.

FALSE A linear transformation is onto is the codomain is equal to the range.

A mapping T : Rn → Rm is one-to-one if each vector in Rn maps onto a unique vector in Rm.

FALSE A mapping is one-to-one if each vectors in Rm is mapped to from a unique vector in Rn.

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

FALSE Again, check properties to show it is a linear transformation.

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 Can only be reduced to one matrix (row reduced echelon form)

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

FALSE For example, [1, 2, 3] and [2, 4, 6] are linearly dependent

An inconsistent system has more than one solution

FALSE Inconsistent system has no solutions!

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

FALSE Rm is the codomain, the range is where we actually land.

IfAisa3×2matrix,thethetransformationx→Axcannot be one-to-one.

FALSE Since the transformation maps from R2 to R3 and 2 < 3 it can be one-to-one but not onto.

The codomain of the transformation x → Ax is the set of all linear combinations of the columns of A.

FALSE The If A is m × n codomain is Rm. The original statement in describing the range.

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

FALSE The domain is R5.

Every elementary row operation is reversible.

TRUE You can reverse multiplying by a constant by multiplying by its inverse. If you add row one to row two and replace row two, then you can subtract row one from row two to get it back.

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

TRUE

The columns of the standard matrix for a linear transformation from Rn to Rm are the images of the columns of the n × n identity matrix.

TRUE

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

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

TRUE

The superposition principle is a physical description of a linear transformation.

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

A homogeneous equation is always consistent.

TRUE - The trivial solution is always a solution.

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

TRUE If they have the same solution set, they both reduce to the same matrix in reduced row echelon form, since row operations are reversible, we can then reverse one set of these to get from one matrix to the other by row operations, thus they are row equivalent.

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 If we take the definition of linear transformation we can derive these and if these are true then they are true for c1, c2 = 1 so the first part of the definition is true, and if v = 0, then the second part if true.

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

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 in in their span.

A linear transformation T : Rn → Rm is completely determined by its effect on columns of the n × n identity matrix.

TRUE The columns on the identity matrix are the basis vectors in Rn. Since every vector can be written as a linear combination of these, and T is a linear transformation, if we know where these columns go, we know everything.

A linear transformation is a special type of function.

TRUE The properties are (i) T(u + v) = T(u) + T(v) and (ii) T(cu) = cT(u).

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

TRUE This is part of the definition of a linear transformation.

Every matrix transformation is a linear transformation

TRUE To actually show this, we would have to show all matrix transformations satisfy the two criterion of linear transformations.

The standard matrix of a linear transformation from R2 to R2 that reflects points through the horizontal axis, the vertical a 0 axis,ortheorignhastheform 0 d ,whereaandd are ±1

TRUE We can check this by checking the images of the basis vectors.

IfAisa3×2matrix,thenthetransformationx→Axcannot map R2 onto R3

TRUE You can not map a space of lower dimension ONTO a space of higher dimension