Vocabulary

Question 1

m by n matrix

Answer 1

Size of the matrix which indicates the number of rows and columns. If m and n are positive integers, an m x n matrix is a rectangular array of numbers with m rows and n columns.

Question 2

row equivalent matrices

Answer 2

Two matrices with the same solution set.

In other words, one matrix can be converted to the other using elementary row operations.

Question 3

echelon form of a matrix

Answer 3

A rectangular matrix is in echelon form (or row echelon form) if it has the following three properties:

1. All nonzero rows are above any rows of all zeroes.

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.

Question 4

reduced row echelon form (RREF) of a matrix

Answer 4

A matrix in echelon form which satisfies the following additional conditions:

1. The leading entry in each nonzero row is 1.

2. Each leading 1 is the only nonzero entry in its column.

Question 5

scalar

Answer 5

A (real) number used to multiply either a vector or a matrix.

Question 6

zero vector

Answer 6

The unique vector, denoted by 0, such that u+0=u for all u. In Rⁿ, 0 is a vector whose entries are all zero.

Ex: v₀ = (0,0,0)

Question 7

linear combination of a set of vectors

Answer 7

A sum of scalar multiples of vectors. The scalars are called weights. Given vectors v₁, v₂, …, v_p in Rⁿ and given scalars c₁, c₂, …, c_p, the vector y defined by y=c₁v₁ + … + c_pv_p is a lin. comb. of v vectors with c weights.

Question 8

pivot position, pivot column

Answer 8

pivot position - In a matrix A it is a location in A that corresponds to a leading # other than zero, or in the reduced echelon form of A, 1.

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

Question 9

the span of a set of vectors (i.e. Span {v₁, v₂, v₃, …, v_n})

Answer 9

The collection of all vectors that can be written in the form c₁v₁+ c₂v₂+ ⋯ + c_pv_p with c₁, ⋯, c_p scalars.

Glossary Deff. – Span {v₁, ⋯ ,v_p }: The set of all linear combinations of v₁, ⋯,v_p. Also, the subspace spanned (or generated) by v₁,⋯,v_p.

The vector space made up of all linear combinations of the set {v₁, v₂, v₃, … , v_n}.

Question 10

homogeneous system of linear equations

Answer 10

A system of linear equations that 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.

Question 11

trivial solution to a homogeneous system

Answer 11

Ax = 0, has at least one solution, namely, x=0(the zero vector in R^m.

Question 12

parametric vector form of a solution to Ax = b.

Answer 12

x = su + tv (s, t ∈ R).

s & t are the weights

u & v are the vectors

 |x1 |    |-1 |  +      |4|

x = |x2| = |2 | + x3 |0|

  |x3| = |2 |  +      |1|

A solution in the form of x = x_p + x_h, where x_h is a linear combination. h = the homogeneous solution and p = the particular solution.

Question 13

set of linearly independent vectors

Answer 13

An indexed set of vectors {v₁, v₂, v₃, …, v_n}) in Rⁿ with a vector equation: c₁v₁+ c₂v₂ + ⋯ + c_pv_p = 0. Where the weights are all zero.

Question 14

set of linearly dependent vectors

Answer 14

If there exist weights c₁, c₂, …, c_p, not all zero, such that:

{v1, v2, v3, …, vn}) in with a vector equation: c1v1+ c2v2 + ⋯ + cpvp = 0.

Question 15

linear transformation (or function or mapping) T from Rⁿ to R^m.

Answer 15

A rule that assigns to each vector x in Rⁿ a vector T(x) in R^m. Must meet the following properties:

1. T(x + y) = T(x) + T(y)
2. T(cx) = cT(x)

Question 16

domain, co-domain, and range of a linear transformation of a Rⁿ to R^m transformation

Answer 16

domain - The set Rⁿ.

co-domain - The set R^m.

range - The set of all images T(x).

Question 17

The image of vector x under the action of a linear transformation T

Answer 17

T(u) = Au = | 3 5 | • | 2 | = | 1 |

               | -1  7|    | -1 |    | -9 |

1 -3| | 5 |

Question 18

contraction, dilation, shear, projection, rotation

Answer 18

contraction – T: R² → R² by T(x) = rx when 0 ≤ r ≤ 1.

dilation – T: R² → R² by T(x) = rx when r > 1

shear: a transformation that changes one portion of a point (x, y) such as x, without changing y.

projection:

x₁ | 1 0 0 | | x₁ | = | x₁ |

x₂ →| 0 1 0 | | x₂ | = | x₂ |

x₃ | 0 0 0| | x₃ | = | 0 |

rotation:

A = | cos σ - sin σ |

   | sin σ      cos σ |

Question 19

standard matrix for a linear transformation

Answer 19

The matrix A such that T(x) = Ax.

Question 20

onto transformation

Answer 20

A mapping T: Rⁿ → R^{<span>m</span>} is said to be onto R^m if each b in R^m is the image of at least one ** x** in Rⁿ.

Many points in the domain can map to the same point in the range.

Question 21

one-to-one transformation

Answer 21

A mapping T: Rⁿ → R^m is said to be one-to-one if each b in R^m is the image of at most one x in Rⁿ. Each point in the domain maps to a unique point in the range.

Question 22

transpose of a matrix

Answer 22

Given an m x n matrix, the transpose of A is the n x m matrix, denoted by AT, whose columns are formed from the corresponding rows of A.

C = | 1 1 1 1 |

   | -3  5 -2  7 |

C^T = | 1 -3 |

    | 1   5  |

    | 1   -2 |

    | 1   7  |

Question 23

invertible matrix

Answer 23

A matrix with the following attributes:

1. An n x n matrix A is said to be invertible if there is an n x n matrix C such that CA = 1 & AC = 1.
2. A matrix A where A^-1 exists.
3. A matrix A that is row equivalent to I_n.
4. A • A^-1 = I_n

Question 24

singular(non-invertable) and non-singular (invertable)

Answer 24

singular - a matrix in Rⁿ (the opposite of each of the items below applies to non-singular)

1. pivots < n (free variables).
2. Ax = 0 has more that one solution.
3. Not invertable.
4. Not row equivalent to I.
5. The columns of A do not span Rⁿ.
6. The columns of A form a linearly dependent set.
7. A^T is singular.
8. T(x) = Ax is not a one-to-one function.
9. T(x) = Ax does not map Rⁿ onto Rⁿ
10. There exist some b in Rⁿ, Ax = b is not consistent.
11. There is not an n by n matrix C such that CA = 1.
12. detA = 0.

Question 25

elementary row operation

Answer 25

There are three: 1. (Replacement) Replace one row by the sum of itself and a multiple of another row. 2. Interchange) Interchange two rows. 3. (Scaling) Multiply all entries in a row by a nonzero constant.

Question 26

elementary matrix

Answer 26

A matrix that is obtained by performing a single elementary row operation on an identity matrix.

Question 27

subspace of Rⁿ

Answer 27

any set H in that has three properties: 1. The zero vector is in H. 2. For each u and v in H, the sum u + v is in H. 3. For each u in H and each scalar c, the vector cu is in H.

Question 28

column space of A

Answer 28

the set Col A of all linear combinations of the columns of A. The colunm b such that Ax = b has a solution.

Question 29

null space of A

Answer 29

the set Nul A of all solutions to the homogeneous equation Ax = 0.

Question 30

basis of a subspace of Rⁿ

Answer 30

A linearly independent set in H that spans H. RREF the matrix and then put in parametric vector form. Must: 1. Be linearly independent 2. Span the subspace

Question 31

coordinate vector of x relative to a basis B

Answer 31

the coordinates of x ralative to the basis ß are the weights c₁, ... , c_p such that x = c₁b₁ + ... + c_pb_p, and the vector in Rp [x]ß = |c₁| |...| |c_p| is called the coordinate vector of x relative to a basis ß

Question 32

dimension of a subspace

Answer 32

the number of bectors in any basis for H.

Question 33

rank of a matrix (denoted by rank A)

Answer 33

the dimension of the column space of A. (# of columns)

Question 34

spanning (or generating) set of a subspace

Answer 34

a sent {v₁, ... , v_p} in H such that H = Span {v₁, ... , v_p}.

Question 35

kernel of a linear transformation (or **null space**)

Answer 35

the set if all **u** in V such that T(**u**) = 0. (the zero vector in W)

Question 36

eigenvalue ŷ of a square matrix

Answer 36

A scalar ŷ(should be upside-down y) is called an **eigenvalue** of A if there is a nontrivial solution **x** of A**x** =**ŷx**; such an **x **is called an eigenvector corresponding to ŷ.

Question 37

eigenvector associated with ŷ(should be upside-down y)

Answer 37

a x such that x is a nontrivial solution of Ax =ŷx A nonzero vector **x** such that A**x** =ŷ**x **for some scalar ŷ.

Question 38

characteristic equation of a square matrix

Answer 38

The scalar equation det(A - ŷI) = 0 ŷ (should be upside-down y)

Question 39

The Invertible Matrix Theorem If A is an invertible matrix.

Answer 39

A^-1 exists

Question 40

The Invertible Matrix Theorem A is/isn't row equivalent to the n x n identity matrix.

Answer 40

A **is** row equivalent to the *n* x *n* identity matrix.

Question 41

The Invertible Matrix Theorem If A is an m x n matix, A has \_\_\_ pivot positions.

Answer 41

A has ***n*** pivot positions.

Question 42

The Invertible Matrix Theorem The equation A**x**=**0** has ___________ solution.

Answer 42

The equation A**x**=**0** has **only the trivial solution.**

Question 43

The Invertible Matrix Theorem The columns of A for a linearly _________ set.

Answer 43

The columns of A for a linearly **independent** set.

Question 44

The Invertible Matrix Theorem The linear transformation x |→ Ax is/isn't one-to-one.

Answer 44

The linear transformation **x |→ Ax** **is** one-to-one.

Question 45

The Invertible Matrix Theorem The equation A**x** = **b** has ________ solution(s) for each **b** in Rⁿ.

Answer 45

The equation **Ax = b** has **at least one** solution for each **b** in Rⁿ.

Question 46

The Invertible Matrix Theorem The columns of A ________ Rⁿ.

Answer 46

The columns of A **span** Rⁿ.

Question 47

The Invertible Matrix Theorem The linear transformations x |→ Ax \_\_\_\_\_\_\_\_ Rⁿ onto Rⁿ.

Answer 47

The linear transformations **x |→ Ax** **maps** Rⁿ onto Rⁿ.

Question 48

The Invertible Matrix Theorem There is an n x n matrix C such that CA = \_\_\_\_.

Answer 48

There is an *n* x *n* matrix** **C such that CA = ***I***.

Question 49

The Invertible Matrix Theorem There is an n x n matrix D such that AD=\_\_\_\_\_\_.

Answer 49

There is an *n* x *n* matrix D such that AD=***I***.

Question 50

The Invertible Matrix Theorem A^T is/isn't an invertible matrix.

Answer 50

A^T **is** an invertible matrix.

Question 51

Determinate of an *n* x *n* matrix A

Answer 51

Computed by a cofactor expansion across any row or down any column.

Question 52

The Invertible Matrix Theorem The columns of A form a ______ of Rⁿ.

Answer 52

The columns of A form a **basis** of Rⁿ.

Question 53

The Invertible Matrix Theorem Col A = R^?.

Answer 53

Col A = R**ⁿ**.

Question 54

The Invertible Matrix Theorem dim Col A = \_\_\_\_\_\_\_

Answer 54

dim Col A = **n**

Question 55

The Invertible Matrix Theorem rank A = \_\_\_\_\_

Answer 55

rank A = **n**

Question 56

The Invertible Matrix Theorem Nul A = {?}

Answer 56

Nul A = {**0**}

Question 57

The Invertible Matrix Theorem dim Nul A = \_\_\_\_\_\_

Answer 57

dim Nul A = **0**

Question 58

The **rank** of matrix A

Answer 58

Denoted by rank A - Is the dimension of the column space of A. Rank is equal to the number of pivot columns in Reduced Row Echelon Form.

Question 59

If A is a 2 x 2 matrix with zero determinant, then one column of A is a ___________ of the other

Answer 59

Multiple

Question 60

If two rows of a 3 x 3 matrix A are the same, then det A = \_\_\_\_\_\_.

Answer 60

0

Question 61

If A is a 3 x 3 matrix, then det 5A _ 5 det A.

Answer 61

Not equal

Question 62

If A and B are n x n matrices, with det A = 2 and det B = 3, then det(A + B) _ 5.

Answer 62

Not eqaul.

Question 63

If A is n x n and det A = 2, then det A³ \_ 6.

Answer 63

Not equal.

Question 64

If B is produced by interchanging two rows of A, then det B _ det A.

Answer 64

Not equal, det B = -det A.

Question 65

If B is produced by multiplying row 3 of A by 5, then det B _ 5 det A.

Answer 65

Equals.

Question 66

If B is formed by adding to one row of A a linear combination of the other rows, then det B _ det A.

Answer 66

Equals.

Question 67

det A^T _ -det A

Answer 67

False. det A^T = det A.

Question 68

det(-A) _ -det A.

Answer 68

Not equals.

Question 69

detA^TA _ 0.

Answer 69

\>=, Greater than or equals to.

Question 70

Any system of n linear equations in n variables can be solved by Cramer's rule - T/F

Answer 70

False

Question 71

If **u** and **v** are in R² and det [**u v**] = 10, then the area of the triangle in the plane with vertices at **0, u,** and **v** is \_\_.

Answer 71

10.

Question 72

If A³ = 0, then det A = \_\_.

Answer 72

0, zero.

Question 73

If A is invertable, then det A^-1 _ det A.

Answer 73

Does not equal. det A^-1 = 1/det A

Question 74

If A is invertable, then (det A)(det A^-1) _ 1.

Answer 74

Equals

Question 75

dimP_n= Where P is a polynomial such x₂+1 = P₂

Answer 75

dimP_n=n+1

Question 76

T is a Linear Transformation if

Answer 76

1. *T*(**u** + **v**) = *T*(**u**) + *T*(**v**) 2. *T*(*c***u**) = *cT*(**u**)

Question 77

basis of column space

Answer 77

The pivot columns of a matrix A form a **basis **for the column space of A.

Question 78

basis of nul space

Answer 78

The pivot columns of the homogeneous solution after RREF.

Question 79

basis of row space

Answer 79

The pivot rows of the modified matrix A after RREF

Question 80

kernel

Answer 80

Same as nul Space

Question 81

dimension of the vector space

Answer 81

The # vectors in the basis or the # of pivots.

Question 82

dimmensions of the nul Space

Question 83

rankA

Answer 83

= dimA

Question 84

The Rank Theorem

Answer 84

For m x n matrix: 1. rank A + dim Nul A = n 2. common dimension = rank A = # of pivots = dim of row space.

Question 85

Let ß = {b₁, ... , b_n} and C = {c₁, ... , c_n} be bases of a vector space V. Then there is a unique *n x n* matric _cP_ß such that

Answer 85

[**x**}_c = _cP_ß [**x**]_ß

Question 86

The unique *n x n* matric _cP_ß is known as:

Answer 86

The **change-of-coordinates matric from ß to C** **such that:** [**x**}_c = _cP_ß [**x**]_ß