Math 355 Chapter 1

Front 1

SYSTEM OF LINEAR EQUATIONS

Back 1

A collection of linear equations with the same variables.

Front 2

LINEAR EQUATION

Back 2

An equation that can be written as c1x1 + c2x2 + ... + cnxn = b with c1-cn being coefficients that are real numbers.

Front 3

SOLUTION TO A SYSTEM OF LINEAR EQUATIONS

Back 3

A collection of values that make the system true.

Front 4

INCONSISTENT SYSTEM

Back 4

A system with no solution.

Front 5

CONSISTENT SYSTEM

Back 5

A system with at least one solution.

Front 6

EQUIVALENT SYSTEMS

Back 6

Two linear systems that have the same solution set.

Front 7

How many solutions can a system have?

Back 7

The only choices are :
1. One
2. Zero
3. Infinitely many

Front 8

ROW ECHELON FORM

Back 8

The form a matrix is in if:
1. All nonzero rows are above zero rows
2. Each leading entry is tot he right of the row above
3. All entires in the column below a leading entry are zeros
NOTE: every matrix has an echelon form, but it is not unique.

Front 9

REDUCED ECHELON FORM

Back 9

The form a matrix is in if:
1. All nonzero rows are above zero rows.
2. Each leading entry is to the right of the row above
3. All entires in the column below a leading entry are zeros.
4. The leading entry in each row is 1.
5. Each leading 1 is the only nonzero entry in its column
NOTE: every matrix has a unique reduced echelon form.

Front 10

MATRIX

Back 10

A rectangular array where information about a linear system can be recorded.

Front 11

COEFFICIENT MATRIX

Back 11

A matrix with only the coefficients of the variables aligned in columns.

Front 12

AUGMENTED MATRIX

Back 12

A matrix that includes coefficients from the left side of the linear system and the constants from the right side of the linear system.

Front 13

SIZE

Back 13

Denotes how many rows and columns a matrix has. The size is read "m by n", where m represents the number of rows and n represents the number of columns a matrix has.

Front 14

ELEMENTARY ROW OPERATIONS

Back 14

1. REPLACEMENT: Replace one row by the sum of itself and a multiple of another row
2. INTERCHANGE: Switch two rows
3. SCALING: Multiply all entries in a row by a nonzero constant
NOTE: Row operations are reversible.

Front 15

ROW EQUIVALENT

Back 15

Two matrices are row equivalent if there is a sequence of elementary row operations that transforms one matrix into the other.

Front 16

EXISTENCE & UNIQUENESS

Back 16

1. Is the solutions consistent; that is, does at least one solution exist?
2. If a solution exists, is it the only one; that is, is the solution unique?

Front 17

LEADING ENTRY

Back 17

The leftmost nonzero entry (in a nonzero row).

Front 18

PIVOT POSITION

Back 18

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

Front 19

PIVOT COLUMN

Back 19

A pivot column is a column of A that contains a pivot position.

Front 20

BASIC/LEADING/PIVOT VARIABLES

Back 20

Variables in A that correspond to pivot columns. All other variables are FREE VARIABLES.

Front 21

GENERAL SOLUTION OF A LINEAR SYSTEM

Back 21

A description of variables in terms of free variables and constants, so that the solution gives an explicit description of all solutions.

Front 22

COLUMN VECTOR

Back 22

A matrix with only with only one column.

Front 23

EQUALITY OF VECTORS IN R2

Back 23

Two vectors in R2 are equal iff their corresponding entries are equal.

Front 24

SCALAR MULTIPLE OF A VECTOR

Back 24

The scalar multiple of vector u by constant c is the vector cu obtained by multiplying each entry in u by c .

Front 25

LINEAR COMBINATION

Back 25

Give vectors v1, v2, ... , vp with weights c1, ... , cp, the linear combination is described as y = c1v1 + c2v2 + ... + cpvp.

Front 26

SPAN

Back 26

If v1, ... , vp are in Rn, then the set of all inear combinations of v1, ... , vp is denoted by Span {v1, ..., vp} and is called the subset of Rn spanned (or generated) by v1, ... , vp. That is, Span {v1, ... , vp} is the collection of all vectors that can be written in the form c1v1 + c2v2 + ... + cpvp with c1, ... , cp scalars.

Front 27

Column Vector

Back 27

A matrix with only one column, also called a vector.

Front 28

Vector Equality

Back 28

Two vectors in R2 are equal iff their corresponding entries are equal.

Front 29

Sum of Vectors

Back 29

Given two vectors u and v in R2, their sum is the vector u + v obtained by adding corresponding entries of u and v.

Front 30

Scalar Multiple

Back 30

Given a vector u and a real number c, the scalar multiple of u by c is the vector cu obtained by multiplying each entry in u by c. The number c in cu is called a scalar.

Front 31

Parallelogram Rule for Addition

Back 31

If u and vin R2 are represented as points in the plane, then u + v corresponds to the fourth vertex of the parallelogram whose other vertices are u, 0, and v.

Front 32

Zero Vector

Back 32

The vector whose entries are all zero, denoted by a boldface 0.

Front 33

Linear Combination

Back 33

Given vectors v1, v2, ... , vp in Rn and given scalars c1, c2, ..., cp the vecctor y defined by y = c1v1 + ... + cpvp is called a linear combination of v1, ... , vp with weights c1, ..., cp.

Front 34

Solutions to Vector Equations

Back 34

A vector equation x1a1 + x2a2 + ... + xnan = b has the same solution set as the linear system whose augmented matrix is [a1 a2 ... an b]. b can be generated by a linear combination of a1, ..., an iff there exists a solution to the linear system corresponding to the augmented matrix.

Front 35

Span

Back 35

If v1, ..., vp are in Rn, then the set of all linear combinations of vi, ..., vp is denoted by Span{v1, ..., vp} and is called the subset of Rn spanned (or generated by v1, ..., vp. That is, Span{v1, ..., vp} is the collection of all vectors that can be written in the form c1v1 + c2v2 + ... + cpvp with c1, ..., cp scalars.

Front 36

BASIS VECTORS

Back 36

(1,0,0), (0,1,0), (0,0,1). The span of these basis vectors generates R3

Front 37

LINEAR COMBINATION

Back 37

If A is an m x n matrix with columns a1, …, an and xεRn, then Ax is the linear combination of the columns of A using the entries of x as the coefficients

Front 38

MATRIX MULTIPLICATION REQUIREMENTS

Back 38

In order to multiply two matrices, the number of columns of the first matrix must be equal to the number of rows in the second matrix, i.e. a 3 x 2 ● 2 x 1 will yield a 3 x 1 matrix product

Front 39

EQUALITY OF SYSTEM NOTATION

Back 39

Linear systems, vector equations, and matrix equations are all interchangeable

Front 40

Ax=b has a solution iff…

Back 40

b in the span of the columns of A

Front 41

Let A be an m x n matrix. The following are equal:

Back 41

1. For every bεRm, the equation Ax=b has a solution
2. Each bεRm is a linear combination of the columns of A
3. The columns of A span Rm
4. The coefficient matrix of A has a pivot in every row. If not, then the matrix is inconsistent

Hint 41

None

Front 42

If A is an m x n matrix; u,vεRn, and c is a scalar, then:

Back 42

1. A(u + v) = Au + Av, 2. A(cu) = c(Au)

Front 43

The homogeneous equation Ax=0 has a nontrivial solution iff…

Back 43

the equation has at least one free variable.

Front 44

PARAMETRIC VECTOR EQUATION OF THE PLANE

Back 44

x = su + tv with s,tεR

Front 45

PARAMETRIC VECTOR FORM

Back 45

Whenever a solution set is described explicitly with vectors, x = p + tv describes the solution set of Ax = b in parametric vector form

Front 46

x = p + tv describes...

Back 46

the equation of the line through p parallel to v

Front 47

Suppose the equation Ax = b is consistent for some given b, and let p be a solution. What can we say about the solution set in relation to the solution set of Ax = 0?

Back 47

The solution set of Ax = b is the set of all vectors of the form w = p + v(h), where v(h) is any solution of the homogeneous equation Ax = 0.

Front 48

Steps for writing a solution set (of a consisten system in parametric vector form:

Back 48

1. Row reduce the augmented matrix to reduced echelon form. 2. Express each basic variable in terms of any free variables appearing in an equation. 3. Write a typical solution x as a vector whose entries depend on the free variables, if any. 4. Decompose x into a linear combination of vectors (with number entries) using the free variables as parameters.

Front 49

LINEAR INDEPENDENCE

Back 49

An indexed set of vectors {v1, … ,vp} εRn is said to be linearly independent if the vector equation x1v1 + … +xpvp = 0 has only the trivial solution.

Front 50

LINEAR DEPENDENCE

Back 50

An indexed set of vectors {v1, … ,vp} εRn is said to be linearly dependent if there exist weights c1, … ,cp not all zero such that quation c1v1 + … +cpvp = 0.

Front 51

The columns of matrix A are linearly independent iff…

Back 51

the equation Ax = 0 has ONLY the trivial solution.

Front 52

A set containing only one vector v in inearly independent iff…

Back 52

v is not the zero vector.

Front 53

A set of two vectors {v1, v2} is linearly independent iff…

Back 53

neither of the vectors is a multiple of the other.

Front 54

In geometric terms, two vectors are linearly dependent iff…

Back 54

they lie on the same line through the origin.

Front 55

An indexed set S = {v1, …, vp} of two or more vectors is linearly dependent iff…

Back 55

AT LEAST one of the vectors in S is a linear combination of the others.

Front 56

What is a warning regarding linearly dependent sets?

Back 56

Not EVERY vector in a linearly dependent set is a linear combination of the preceding vectors. A vector in a linearly dependent set may fail to be a linear combination fof the other vectors.

Front 57

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

Back 57

linearly dependent. Another way to say this is any set {v1, …, vp}εRn is linearly dependent if p > n.

Front 58

If a set S= {v1, …, vp}εRn contains the zero vector, then the set is…

Back 58

linearly dependent. Another way to say this is any set {v1, …, vp}εRn is linearly dependent if p > n.

Front 59

TRANSFORMATION/FUNCTION/MAPPING

Back 59

Defined as T:Rn→Rm that assigns to each vecto xεRn a vector T(x)εRm

Front 60

DOMAIN OF T

Back 60

Rn

Front 61

CODOMAIN OF T

Back 61

Rm

Front 62

IMAGE

Back 62

For xεRn, the vector T(x)εRm is the IMAGE of x under the action of T

Front 63

RANGE

Back 63

The RANGE of T is the set of all images T(x), or the set of all linear combinations of the columns of A

Front 64

A transformation T is linear iff:

Back 64

1. T(u + v) = T(u) + T(v) for all u, v in the domain of T 2. T(cu) = cT(u) for all u and all scalars c 3. T(0) = 0

Front 65

Every matrix transformation is also…

Back 65

a linear transformation

Front 66

TRANSFORMATION/FUNCTION/MAPPING

Back 66

Defined as T:Rn→Rm that assigns to each vecto xεRn a vector T(x)εRm

Front 67

DOMAIN OF T

Back 67

Rn

Front 68

CODOMAIN OF T

Back 68

Rm

Front 69

IMAGE

Back 69

For xεRn, the vector T(x)εRm is the IMAGE of x under the action of T

Front 70

RANGE

Back 70

The RANGE of T is the set of all images T(x), or the set of all linear combinations of the columns of A

Front 71

A transformation T is linear iff:

Back 71

1. T(u + v) = T(u) + T(v) for all u, v in the domain of T 2. T(cu) = cT(u) for all u and all scalars c 3. T(0) = 0

Front 72

Every matrix transformation is also…

Back 72

a linear transformation

Front 73

STANDARD MATRIX FOR A LINEAR TRANSFORMATION T

Back 73

Let T: Rn→Rm be a linear transformation. Then there exists a unique matrix A such that T(x) = Ax for all xεRn. A is the standard matrix, or the m x n matrix whose jth column is the vector T(ej), where ej is the jth column of the identity matrix ε Rn. A = [T(e1) ... T(en)]

Front 74

What is the conceptual difference between a linear transformation and a matrix transformation?

Back 74

The term linear transformation focuses on a property of a mapping, while matrix transformation describes how such a mapping is implemented

Front 75

ONTO

Back 75

A mapping T:Rn→Rm is said to be onto Rm if each bεRm is the image of at least one xεRn, or equivalently, when the range of T is all of the codomain Rm. "Does T map Rn onto Rm?" is an existence question!

Front 76

ONE-TO-ONE

Back 76

A mapping T:Rn→Rm is said to be one-to-one if each bεRm is the image of at most one xεRn, or equivalently, T is one-to-one if for each bεRm, the quation T(x) = b has either a unique solution or none at all. "Is T one-to-one" is a uniqueness question!

Front 77

Let T: Rn→Rm be a linear transformation. Then T is oneto-one iff…

Back 77

the equation T(x) = 0 has only the trivial solution.

Front 78

Let T: Rn→Rm be a linear transformation and let A be the standard matrix for T. Then:

Back 78

1. T maps Rn onto Rm iff the columns of A span Rm 2. T is one-to-one iff the columns of A are linearly independent