Math 240 1.1 - 2.2

Definition: Pivot Position

A pivot position in matrix A is a location in A that corresponds to a leading 1 in the reduced echelon form of A. a pivot column is a column of A that contains a pivot position

True or False: 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, The statement is only true when there exists some vector p such that Ap = b

True or False: If A and B are 2 x 2 with columns a1a2 and b1b2, respectively, then AB = [a1b1 a2b2]

False, a1, a2, b1, and b2 are columns not rows

True or False: An inconsistent system has more than 1 solution.

False, an inconsistent system has no solutions

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

False, an m x n matrix has m rows and n columns

True or False: the equation Ax = b is consistent if the augmented matrix [A b] has a pivot position in every row

False, we cannot determine that using theorem 4 on an augmented matrix

Theorem 4

Let A be an m x n matrix. Then the following statements are logically equivalent. That is, for a particular A, either they are all true statements or they are all false. a. For each *b* in R^m, the equation A*x* = *b* has a solution. b. Each *b* in R^m is a linear combination of the columns of A. c. The columns of A span R^m. d. A has a pivot position in every row.

True or False: Two fundamental questions about a linear system involve existence and uniqueness

True The two fundamental questions about a linear equation are: (i) Is the system consistent, that is, does at least one solution exists? (ii) If a solution exists, is it the only one, that is, is the solution unique.

True or False: 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, basic variables correspond to the columns that have a leading 1's (pivot columns).

True or False: if v1, . . ., v4 are in R^4 and v3 = 0 then {v1, v2, v3, v4} is linearly dependent

True, any zero vector within a set means the set is linearly dependent

zero vector

the vector whose entries are all zero

A transformation (or mapping) T is linear if:

(i) T (u + v) = T(u) + T(v) for all u, v in the domain of T; (ii) T (cu) = cT(u) for all u and all scalars c

Row Echelon Form properties

1. All nonzero rows are above any rows of all zeros 2. Each leading entry of a row is in a column to the right of the leading entry of the row above it 3. All entries in a column below a leading entry are zeroes

Suppose T: R^5 -> R^2 and T(x) = Ax for some matrix A and for each x in R^5. How many rows and columns does A have?

2 rows, 5 columns

A matrix with only one column

A column vector or just vector

One-to-one

A mapping T:R^n->R^m is said to be one-to-one if each b in R^m is the image of at most one x in R^n.

Sets of 1 or 2 vectors linear dependence

A set of two vectors {v1, v2} is linearly dependent if at least one of the vectors is a multiple of the other. The set is linearly independent if and only if neither of the vectors is a multiple of the other.

linear combination

A sum of scalar multiples of vectors. The scalars are called the weights.

Homogeneous Linear System

A system of linear equations is homogeneous if it can be written in the form Ax = 0 where A is an m x n matrix and 0 is the zero vector in R^m. Such a system always has at least one solution (x = 0).

Theorem 7: Characterization of Linearly Dependent Sets

An indexed set S = {v1, . . ., vp} of two or more vectors is linearly dependent if and only if at least one of the vectors in S is a linear combination of the others. In fact, if S is linearly dependent and v1 != 0 , then some vj (with j > 1) is a linear combination of the preceding vectors, v1, . . ., vj-1

Suppose a system of linear equations has a 3 x 5 augmented matrix whose fifth column is a pivot column. Is the system consistent? Why or why not?

Definition:( Existence and Uniqueness Theorem) A linear system is consistent if and only if the right most column of the augmented matrix is not a pivot column. Hence, such a system must be inconsistent.

Restate the last sentence in Theorem 2 using the concept of pivot columns: "If a linear system is consistent, then the solution is unique if and only if ______"

Each column in the matrix is a pivot column.

What would you have to know about the pivot columns in an augmented matrix in order to know that the linear system is consistent and has a unique solution?

Each columns would have to be a pivot column except for the rightmost column.

Theorem 1: Uniqueness of the Reduced Echelon Form

Each matrix is row equivalent to one and only one reduced echelon matrix.

True or False: If A is a 3 x 2 matrix, then the transformation x --> Ax cannot be one-to-one

False

True or False: the echelon form of a matrix is unique

False

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

False, By theorem 1, each matrix is row equivalent to one and only one reduced echelon matrix. Therefore, pivot position of a matrix is unique.

True or False: The solution set of a linear system involving variables x_1, ..., x_n is a list of numbers (s_1, ..., s_n) that makes each equation in the system a true statement when the values s_1, ..., s_n are substituted for x_1, ..., x_n respectively.

False, The description given has been applied to a single solution whereas the solution set consists of all possible solutions. Only in special cases does the solution set consist of exactly one solution.

True or False: The equation Ax = 0 gives an explicit description of its solution set

False, The equation Ax = 0 gives an implicit description of the solution set. Solving this equation amounts to finding an explicit description of the plane.

True or False: if v1, . . ., v4 are in R^4 and v3 is not a linear combination of v1, v2, v4 then {v1, v2, v3, v4} is linearly dependent

False, another vector could be a linear combination.

True or False: The row reduction algorithm applies only to augmented matrices for a linear system.

False, any matrix can be reduced with the row reduction algorithm

True or 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 has one reduced echelon form.

True or False: if S is a linearly dependent set, then each vector is a linear combination of the other vectors in S

False, if S is linearly dependent then there exists some vector that is a linear combination of some other vectors in S

True or False: if v1 and v2 are in R^4 and v2 is not a scalar multiple of v1 then {v1, v2} is linearly independent

False, if one of the two vectors is a 0 vector, the set is linearly dependent

True or 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, if the 5 was in the last row it would be inconsistent.

True or False: Two matrices are row equivalent if they have the same number of rows

False, row equivalence requires there to be a sequence of operations that can transform one matrix into the other

True or False: the homogeneous equation Ax = 0 has the trivial solution if and only if the equation has at least one free variable

False, that statement applies to the nontrivial solution

True or False: if A is a 3 x 5 matrix and T is a transformation defined by T(x) = Ax then the domain of T is R^3

False, the codomain is R^3 and the domain is R^5

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

False, the columns are linearly independent if the equation Ax = 0 has only the trivial solution

True or False: When two linear transformation are performed one after another, the combined effect may not always be a linear transformation

False, the composition of two linear transformations is still a linear transformation

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

False, the equation x = p + tv describes a line passing through p parallel to v.

True or False: Another notation for the vector [-4] [3] is [-4 3]

False, the first vector is 2x1, the second is 1x2

True or False: Whenever a system has free variables, the solution set contains many solutions.

False, the solution set could have no solutions

Asking whether a vector b is in Span{v1, ..., vp} amounts to asking whether the vector equation X1v1 + X2v2 + ... + Xpvp = b

Has a solution, or equivalently, asking whether the linear system with augmented matrix [v1 . . . vp b] has a solution

Theorem 3

If A is an m x n matrix, with columns a1, ..., an, and if b is in R^m, the matrix equation Ax = b has the same solution set as the vector equation x1a1 + x2a2 + ... + xnan = b Which, in turn, has the same solution set as the system of linear equations whose augmented matrix is [a1 a2 . . . an b]

Theorem 5

If A is an mxn matrix, u and v are vectors in R^n, and c is a scalar, then: a. A(u+v) = Au+Av b. A(cu) = c(Au)

Reduced Row Echelon Form properties

If a matrix is in row echelon form as well as: 1. The leading entry in each nonzero row is 1 2. Each leading 1 is the only nonzero entry in its column

Theorem 9

If a set S = {v1,...,vp} in Rn contains the zero vector, then the set is linearly dependent.

Theorem 8

If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set {v1, ..., vp} in R^n is linearly dependent if p > n.

Suppose the coefficient matrix of a linear system of three equations in three variables has a pivot in each column. Explain why the system has a unique solution.

If there are three variables and three columns then there are no free variables, so the solution is unique.

Explain why a vector w is in Span{u,v} if and only if {u,v,w} is linearly dependent

If w is a linear combination of u and v, then {u, v, w} is linearly dependent, by Theorem 7.

Theorem 11

Let T : Rn -> Rm be a linear transformation. Then T is one-to-one if and only if the equation T(x) = 0 has only the trivial solution.

Theorem 12

Let T: R^n -> R^m be a linear transformation and let A be the standard matrix for T. Then: a. T maps R^n onto R^m if and only if the columns of A span R^m; b. T is one-to-one if and only if the columns of A are linearly independent

Theorem 10

Let T: R^n -> R^m be a linear transformation. Then there exists a unique matrix A such that T(x) = Ax for all x in R^n. IN fact, A is the m x n matrix whose jth column is the vector T(ej) where ej is the jth column of the identity matrix in R^n : A = [ T(e1) ... T(en)]

if V1, ..., Vp are in R^n, then the set of all linear combinations of V1, ..., Vp is denoted by Span {V1, ..., Vp} and is called the subset of R^n spanned ( or generated) by V1, ..., Vp. Meaning...

Span {V1, ..., Vp} is the collection of all vectors that can be written in the form C1V1 + C2V2 + ... + CpVp with C1, ..., Cp scalars

Theorem 6

Suppose the equation Ax=b is consistent for some given b, and let p be a solution. Then 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 homogeneous equation Ax=0.

A transformation (or function, or mapping)

T from R^n to R^m is a rule that assigns to each vector x in R^n a vector T(x) in R^m

If T is a linear transformation, then

T(0) = 0 and T(cu + dv) = cT(u) + dT(v) for all vectors u, v in the domain of T and all scalars c, d.

Linear Independence of Matrix Columns

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

Existence of Solutions

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

standard matrix for a linear transformation

The matrix A such that T(x) = Ax for all <x> in the domain of T. A = [T(e1) . . . T(en)]

Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Explain why the system is consistent.

The system is consistent because the rightmost column of the augmented matrix is not a pivot column. This means that the reduced echelon form of the augmented matrix for this system cannot have a row that looks like [0 ... 0 b] where b is nonzero According to the Existence and Uniqueness Theorem, A system is consistent when the augmented matrix does not have a pivot in the rightmost column or its echelon form has no row of the form [0 ... 0 b] where b is nonzero Therefore, this system is consistent.

Suppose a 3 x 5 coefficient matrix for a system has three pivot columns. Is the system consistent? Why or why not?

The system is consistent because there is a pivot in every row

Let A = [1 0] [0 -1] Give a geometric description of the transformation x [--> Ax

The transformation is a reflection over the x axis

True or False: A general solution of a system is an explicit description of all solutions of the system

True

True or False: A homogeneous equation is always consistent

True

True or False: A linear transformation T: R^n -> R^m is completely determined by its effect on the columns of the n x n identity matrix

True

True or False: A linear transformation is a special type of function

True

True or False: 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

True or False: Elementary row operations on an augmented matrix never change the solution set of the associated linear matrix.

True

True or False: Finding a parametric description of the solution set of a linear system is the same as solving the system

True

True or False: If the columns of an m x n matrix A span R^m, then the equation Ax = b is consistent for each b in R^m

True

True or False: Reducing a matrix to echelon form is called the forward phase of the row reduction process

True

True or False: The columns of a standard matrix for a linear transformation from R^n to R^m are the images of the columns of the n x n identity matrix

True

True or False: 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

True or False: the points in the plane corresponding to [-2] [5] and [-5] [2] lie on a line through the origin

True

True or False: Every elementary row operation is reversible.

True

True or False: Two linear systems are equivalent if they have the same solution set.

True, each solution of the first system is a solution of the second system and each solution of the second system is a solution of the first system.

True or False: A mapping T: R^n -> R^m is onto R^m if every vector x in R^n maps to some vector in R^m

True, if every vector maps then the definition of onto is satisfied

True or False: if T: R^2 -> R^2 rotates vectors about the origin through the angle x, then T is a linear transformation

True, the transformation is a matrix transformation and therefore a linear transformation

True or False: An example of a linear combination of vectors v1 and v2 is the vector (1/2)v1

True, the weights are 1/2 and 0 for v1 and v2 respectively

True or False: the columns of any 4 x 5 matrix are linearly dependent

True, there are more columns than rows so there will be a free variable

True or False: if v1, . . ., v4 are in R^4 and {v1, v2, v3} is linearly dependent then {v1, v2, v3, v4} is also linearly dependent

True, there are still linear combinations

True or False: if A is a 3 x 2 matrix, then the transformation x --> Ax cannot map R^2 onto R^3

True, there cannot be an onto mapping from a lower dimension to a higher dimension

True or False: if x and y are linearly independent, and if {x, y, z} is linearly independent, then z is in Span{x, y}

True, this implies that z is a linear combination of x and y

True or False: if v1 . . . v4 are in R^4 and v3 = 2v1 + v2, then {v1, v2, v3, v4} is linearly dependent

True, v3 being a linear combination of v1 and v2 means that the set is linearly dependent

Prove that u + v = v + u for any u and v in R^n

Vectors in R^n follow algebraic principles, meaning changing the order of the vectors does not change the sum

A vector equation X1A1 + X2A2 + ... + XnAn = b has the same solution set as the linear system whose augmented matrix is

[A1 A2 ... An b] In particular, b can be generated by a linear combination of A1, ..., An if and only if there exists a solution to the linear system corresponding to the matrix

Theorem 2: Existence and Uniqueness

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. If a linear system is consistent, then the solution set contains either (i) a unique solution, when there are no free variables, or (ii) infinitely many solutions, when there is at least one free variable.

Onto

a mapping T: R^n-->R^m is said to be onto R^m if each b in R^m is the image of at least one x in R^n

linearly dependent

an indexed set {v1, .. , vp} with the property that there exist weights c1, ... , cp, not all zero, such that c1v1 + c2v2 + ... + cpvp = 0. That is, the vector equation c1v1 + c1v2 + ... + cpvp = 0 has a non trivial solution

linearly independent

an indexed set {v1, ..., vp} with the property that the vector equation c1v1 + c2v2 + ... + cnvn = 0 has only the trivial solution

A system of linear equations with fewer equations than unknowns is sometimes called an underdetermined system. Suppose that such a system happens to be consistent. Explain why the system has to have infinite solutions.

if it has fewer equations than unknowns that is the equivalent of saying there are fewer rows than columns. If there are fewer rows than columns then there will be free variables, so there are infinite solutions.

Suppose that {v1, v2, v3} is a linearly dependent set of vectors in R^n and v4 is a vector in R^n. Show that {v1, v2, v3, v4} is also a linearly dependent set.

it is implied that there exists scalars c1, c2, c3, not all zero, such that c1v1 + c2v2 + c3v3 = 0 adding 0v4 = 0 to both sides of this equation results in c1v1 + c2v2 + c3v3 + 0v4 = 0 Since c1, c2, c3, and 0 are not all 0, the set {v1, v2, v3, v4} satisfies the definition of a linearly dependent set

Let T: R^n -> R^m be a linear transformation and let A be the standard matrix for T. Complete the following statement to make it true: "T is one-to-one if and only if A has _____ pivot columns." explain why the statement is true

n, because there must be a pivot column in every row

When does the homogeneous equation Ax = 0 have a nontrivial solution?

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

Suppose a 4 x 7 coefficient matrix for a system of equations has 4 pivots. Is the system consistent? If the system is consistent, how many solutions are there?

the system is consistent because there is a pivot in each row. there are infinite solutions because there will be three free variables.

Find the general solution of the system x_1 - 2x_2 - x_3 + 3x_4 = 0 -2x_1 + 4x_2 + 5x_3 - 5x_4 = 3 3x_1 - 6x_2 - 6x_3 + 8x_4 = 2

the system is inconsistent

Two vectors in R^n are equal if and only if...

their corresponding components are equal.

Find the general solution of the linear system whose augmented matrix is [1 -3 -5 0] [0 1 -1 -1]

{ x_1 = 3x_2 + 5x_3 x_2 = -1 + x_3 x_3 is free }