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MTH 227 - Linear Algebra > Vocabulary > Flashcards

Vocabulary Flashcards

1
Q

m by n matrix

A

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.

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2
Q

row equivalent matrices

A

Two matrices with the same solution set.

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

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3
Q

echelon form of a matrix

A

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.

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4
Q

reduced row echelon form (RREF) of a matrix

A

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.

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5
Q

scalar

A

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

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6
Q

zero vector

A

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)

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7
Q

linear combination of a set of vectors

A

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

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8
Q

pivot position, pivot column

A

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.

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9
Q

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

A

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

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

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

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10
Q

homogeneous system of linear equations

A

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 Rm.

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11
Q

trivial solution to a homogeneous system

A

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

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12
Q

parametric vector form of a solution to Ax = b.

A

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 = xp + xh, where xh is a linear combination. h = the homogeneous solution and p = the particular solution.

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13
Q

set of linearly independent vectors

A

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

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14
Q

set of linearly dependent vectors

A

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

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

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15
Q

linear transformation (or function or mapping) T from Rn to Rm.

A

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

  1. T(x + y) = T(x) + T(y)
  2. T(cx) = cT(x)
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16
Q

domain, co-domain, and range of a linear transformation of a Rn to Rm transformation

A

domain - The set Rn.

co-domain - The set Rm.

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

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17
Q

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

A

T(u) = Au = | 3 5 | • | 2 | = | 1 |

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

1 -3| | 5 |

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18
Q

contraction, dilation, shear, projection, rotation

A

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 |
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19
Q

standard matrix for a linear transformation

A

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

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20
Q

onto transformation

A

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

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

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21
Q

one-to-one transformation

A

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

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22
Q

transpose of a matrix

A

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 |

CT = | 1 -3 |

    | 1   5  |

    | 1   -2 |

    | 1   7  |
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23
Q

invertible matrix

A

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 In.
  4. A • A-1 = In
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24
Q

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

A

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. AT 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.
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25
Q

elementary row operation

A

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.
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26
Q

elementary matrix

A

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

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27
Q

subspace of Rn

A

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.
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28
Q

column space of A

A

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

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29
Q

null space of A

A

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

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30
Q

basis of a subspace of Rn

A

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
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31
Q

coordinate vector of x relative to a basis B

A

the coordinates of x ralative to the basis ß are the weights c1, … , cp such that x = c1b1 + … + cpbp, and the vector in Rp

[x]ß = |c1|

      |...|

      |c<sub>p</sub>|

is called the coordinate vector of x relative to a basis ß

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32
Q

dimension of a subspace

A

the number of bectors in any basis for H.

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33
Q

rank of a matrix (denoted by rank A)

A

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

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34
Q

spanning (or generating) set of a subspace

A

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

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35
Q

kernel of a linear transformation (or null space)

A

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

36
Q

eigenvalue ŷ of a square matrix

A

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

37
Q

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

A

a x such that x is a nontrivial solution of Ax =ŷx

A nonzero vector x such that Ax =ŷ**x **for some scalar ŷ.

38
Q

characteristic equation of a square matrix

A

The scalar equation det(A - ŷI) = 0

ŷ (should be upside-down y)

39
Q

The Invertible Matrix Theorem

If A is an invertible matrix.

A

A-1 exists

40
Q

The Invertible Matrix Theorem

A is/isn’t row equivalent to the n x n identity matrix.

A

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

41
Q

The Invertible Matrix Theorem

If A is an m x n matix, A has ___ pivot positions.

A

A has n pivot positions.

42
Q

The Invertible Matrix Theorem

The equation Ax=0 has ___________ solution.

A

The equation Ax=0 has only the trivial solution.

43
Q

The Invertible Matrix Theorem

The columns of A for a linearly _________ set.

A

The columns of A for a linearly independent set.

44
Q

The Invertible Matrix Theorem

The linear transformation x |→ Ax is/isn’t one-to-one.

A

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

45
Q

The Invertible Matrix Theorem

The equation Ax = b has ________ solution(s) for each b in Rn.

A

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

46
Q

The Invertible Matrix Theorem

The columns of A ________ Rn.

A

The columns of A span Rn.

47
Q

The Invertible Matrix Theorem

The linear transformations x |→ Ax ________ Rn onto Rn.

A

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

48
Q

The Invertible Matrix Theorem

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

A

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

49
Q

The Invertible Matrix Theorem

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

A

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

50
Q

The Invertible Matrix Theorem

AT is/isn’t an invertible matrix.

A

AT is an invertible matrix.

51
Q

Determinate of an n x n matrix A

A

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

52
Q

The Invertible Matrix Theorem

The columns of A form a ______ of Rn.

A

The columns of A form a basis of Rn.

53
Q

The Invertible Matrix Theorem

Col A = R<span>?</span>.

A

Col A = Rn.

54
Q

The Invertible Matrix Theorem

dim Col A = _______

A

dim Col A = n

55
Q

The Invertible Matrix Theorem

rank A = _____

A

rank A = n

56
Q

The Invertible Matrix Theorem

Nul A = {?}

A

Nul A = {0}

57
Q

The Invertible Matrix Theorem

dim Nul A = ______

A

dim Nul A = 0

58
Q

The rank of matrix A

A

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.

59
Q

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

A

Multiple

60
Q

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

A

0

61
Q

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

A

Not equal

62
Q

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

A

Not eqaul.

63
Q

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

A

Not equal.

64
Q

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

A

Not equal, det B = -det A.

65
Q

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

A

Equals.

66
Q

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

A

Equals.

67
Q

det AT _ -det A

A

False. det AT = det A.

68
Q

det(-A) _ -det A.

A

Not equals.

69
Q

detATA _ 0.

A

>=, Greater than or equals to.

70
Q

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

A

False

71
Q

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 __.

A

10.

72
Q

If A3 = 0, then det A = __.

A

0, zero.

73
Q

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

A

Does not equal.

det A-1 = 1/det A

74
Q

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

A

Equals

75
Q

dimPn=

Where P is a polynomial such x2+1 = P2

A

dimPn=n+1

76
Q

T is a Linear Transformation if

A
  1. T(u + v) = T(u) + T(v)
  2. T(cu) = cT(u)
77
Q

basis of column space

A

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

78
Q

basis of nul space

A

The pivot columns of the homogeneous solution after RREF.

79
Q

basis of row space

A

The pivot rows of the modified matrix A after RREF

80
Q

kernel

A

Same as nul Space

81
Q

dimension of the vector space

A

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

82
Q

dimmensions of the nul Space

A
83
Q

rankA

A

= dimA

84
Q

The Rank Theorem

A

For m x n matrix:

  1. rank A + dim Nul A = n
  2. common dimension = rank A = # of pivots = dim of row space.
85
Q

Let ß = {b1, … , bn} and C = {c1, … , cn} be bases of a vector space V. Then there is a unique n x n matric cPß such that

A

[x}c = cPß [x]ß

86
Q

The unique n x n matric cPß is known as:

A

The change-of-coordinates matric from ß to C such that:

[x}c = cPß [x]ß

MTH 227 - Linear Algebra (2 decks)

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