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Mathematics > Vectors and Matrices > Flashcards

Vectors and Matrices Flashcards

1
Q

What is the vector AB

A

Let A and B be two points in R^3. The vector AB is the segment with starting point A and ending point B.

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2
Q

What is the defenition of a position vector OA

A

Let A = (xA, yA, zA) be a point in R
3. The position vector OA is
defined by (xA)i+(yA)j+(zA)k

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3
Q

Addition and multiplication by real scalars in R^n are defined as…

A

i) for all x = (x1, x2, … , xn) and y = (y1, y2, … , yn),
x + y = (x1 + y1, x2 + y2, … , xn + yn)

ii) for all x = (x1, x2, … , xn) and λ ∈ R,
λx = (λx1, λx2, … , λxn)

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4
Q

What is the sum of the vectors OA + OB

A

It is s the vector OC which represents the diagonal of the parallelogram constructed on OA and OB.

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5
Q

Let A = (xA, yA, zA) and B = (xB, yB, zB) be two points in R^3. The
vector AB is the sum…

A

OB - OA = (xB − xA)i + (yB − yA)j + (zB − zA)k = AB

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6
Q

Addition and Multiplication properties in R

A

1) Addition in R is commutative

2) Addition in R is associative

3) 0 is the identity for the addition

4) −x is the additive inverse of x

5) multiplication in R is commutative

6) multiplication in R associative

7) 1 is the identity for the multiplication

8) 1/x is the multiplicative inverse of x

9) distributive property

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7
Q

Addition and Scalar Multiplication in R^n

A

1) Addition in R is commutative

2) Addition in R is associative

3) 0 is the identity for the addition

4) −x is the additive inverse of x

5) multiplication in R associative

6) 1 is the identity for the multiplication

7) distributive property

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8
Q

What is a Vector space?

A

A set V with two operations (addition and multiplication by scalars) is called vector space.

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9
Q

What is the standard vector ei.

A

Let i = 1, … , n. The standard vector ei is the column vector with the
i-th entry equal to 1 and all the others equal to 0.

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10
Q

How can any vector v in R^n be represented in terms of the standard basis vectors?

A

Every vector v in R^n can be written as a unique linear combination of the standard vectors, i.e., there exists a unique choice of vi ∈ R, i = 1, … , n, such that
v = sum i=1 to n viei

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11
Q

How is the length (or norm or module) of a vector v in R^n defined?

A

|v| = sqrt(sum i=1 to n (vi)^2)

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12
Q

What is the Unit vector?

A

By multiplying v by the scalar 1/|v| we obtain a unit vector, i.e., a vector with
norm 1

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13
Q

What does the norm of a vector v represent geometrically in R^3 when v is described in terms of the standard basis vectors i,j,k, and corresponds to the point P.

A

|v| is the the length of the segment OP

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14
Q

What is the vector equation for a line l?

A

The equation r = p + λu is called the vector equation for l

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15
Q

What are the parametric equations of a line l

A

x = p1 + λu1
y = p2 + λu2
z = p3 + λu3

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16
Q

What is the cartesian equation of a line l

A

( x - p1 ) / u1
(y - p2 ) / u2
(z - p3) / u3
are all equal too each other as well as λ

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17
Q

What are the closure properties of vector addition and scalar multiplication in a vector space V?

A

For all v, v1, v2 ∈ V and all α ∈ R,
v1 + v2 ∈ V,
αv ∈ V

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18
Q

How is a sub-vector space of R^2 defined with respect to the standard basis vectors i and j, and what geometric object does this set correspond to?

A

The set, V = {xi + yj : x, y ∈ R},
is a sub-vector space of R^2

V is the set of all linear combinations of i and j and it is denoted by Span(i,j). Geometrically speaking it is the (x, y)-plane.

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19
Q

How is the scalar product (also known as the dot product) of two vectors defined?

A

u · v = |u||v|cos θ if u ̸= 0, v ̸= 0
where θ is the angle between u and v

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20
Q

When are two vectors considered to be orthogonal, and what are the conditions that characterize this relationship?

A

Two vectors u and v are considered orthogonal if their scalar product (dot product) is zero, denoted as
u ⋅ v = 0. Vectors are orthogonal if either one or both of them is the zero vector, or they are perpendicular to each other.

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21
Q

What is the formula for calculating the scalar product of two vectors in coordinate form in three-dimensional space?

A

If u and v are vectors in three-dimensional space with coordinates
𝑢 = (𝑢1,𝑢2,𝑢3) and 𝑣=(𝑣1,𝑣2,𝑣3) ,respectively, then the scalar product u ⋅ v is given by the formula: 𝑢⋅𝑣 = 𝑢1𝑣1 + 𝑢2𝑣2 + 𝑢3𝑣3.

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22
Q

What are the key properties of the scalar product for vectors in Euclidean space?

A

1) Commutative

2) Distributive over vector addition

3) Distributive over vector addition (other order)

4) Compatible with scalar multiplication

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23
Q

What is the Cauchy-Schwarz Inequality for vectors in R^3?

A

Let u and v be two vectors in R^3
.The following inequality holds

|u · v| ≤ |u||v|.

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24
Q

What is the Triangle inequality?

A

Let u and v be two vectors in R^3
. The following inequality holds:

|u + v| ≤ |u| + |v|.

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25
Q

What is the vector equation of a plane?

A

r · n = p · n

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26
Q

What is the cartesian equation for a plane?

A

ax + by + cz = d

where d = n1p1 + n2p2 + n3p3

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27
Q

How is the distance from a point to a plane determined?

A

The distance D between point Q and plane Π is given by

D = | q⋅n−d |
|n|

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28
Q

how can you find the position vector of the point on the plane that is closest to a given point?

A

The position vector of the point on the plane Π that is closest to point
Q is given by
q’ = q - n(q⋅n−d)
|n|^2

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29
Q

How is the vector product (also known as the cross product) of two vectors defined in three-dimensional space?

A

Given two vectors 𝑢=(𝑢1,𝑢2𝑢3) and 𝑣=(𝑣1,𝑣2,𝑣3) in three-dimensional space, the vector product
u × v is defined as the vector:

u × v = (u2v3 − u3v2)i − (u1v3 − u3v1)j + (u1v2 − u2v1)k

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30
Q

What is the length of the vector product u×v and how is it related to the vectors u and v?

A

Given vectors u and v, the vector product u × v is orthogonal to
both u and v, and its length |u × v| satisfies:

|u × v| is equal to the area of the
parallelogram

u × v | = |u||v|sin θ if u ̸= 0, v ̸= 0

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31
Q

How can you determine the distance from a point X with position vector x to a line
l with a given vector equation?

A

|MX| = |v|sin θ = |u × v| = |u × (x-p)|
|u| |u|

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32
Q

How do you determine the shortest distance between two skew (non-intersecting and non-parallel) lines in R^3

A

The shortest distance d between two skew lines l1 with vector equation 𝑟=𝑝+𝜆𝑢 and l2 with vector equation 𝑟=𝑞+𝜇𝑣 can be found using the formula:

d = |(q - p)·(u x v)|
|u x v|

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33
Q

What is an inconsistent system?

A

A system with no solution is called inconsistent

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34
Q

What is a consistent system?

A

A system with at least one solution
is called consistent

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35
Q

What is the solution set?

A

The set of all solutions of a system is called its solution set, which may be empty if the system is inconsistent.

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36
Q

When are two systems of linear equations, each with m equations and n unknowns, considered to be equivalent?

A

Two m×n systems of linear equations are said to be equivalent if they share the exact same solution set.

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37
Q

What operations can be performed on a system of linear equations without changing its solution set?

A

(i) interchanging two equations;

(ii) multiplying an equation by a non-zero scalar;

(iii) adding a multiple of one equation to another.

38
Q

What is the augmented matrix of a linear system, and how is it structured?

A

The augmented matrix of an
m×n linear system is the array formed by including the coefficient matrix of the system along with the constants from the equations on the right-hand side.

39
Q

What are the elementary row operation?

A

There are three types of elementary row operations that can be applied to an augmented matrix to solve a system of equations:

Type I: Interchanging two rows.

Type II: Multiplying a row by a non-zero scalar.

Type III: Adding a multiple of one row to another row.

40
Q

What are the conditions that define a matrix as being in row echelon form?

A

(i) All zero rows (consisting entirely of zeros) are at the bottom.

(ii) The first non-zero entry from the left in each nonzero row is a 1, called the leading
1 for that row.

(iii) Each leading 1 is to the right of all leading 1’s in the rows above it.
A row echelon matrix is said to be in reduced row echelon form if, in addition it satisfies
the following condition:

(iv) Each leading 1 is the only nonzero entry in its column

Roughly speaking, a matrix is in row echelon form if the leading 1’s form an echelon (that
is, a ‘steplike’) pattern.

41
Q

What are leading and free variables?

A

The variables corresponding to the leading 1’s of the augmented matrix in row echelon form will be referred to as the leading variables, the remaining ones as the free variables.

42
Q

How to perform Gaussian algorithm

A

Step 1) If the matrix consists entirely of zeros, stop — it is already in row echelon form.

Step 2) Otherwise, find the first column from the left containing a non-zero entry (call it
a), and move the row containing that entry to the top position.

Step 3) Now multiply that row by 1/a to create a leading 1.

Step 4) By subtracting multiples of that row from rows below it, make each entry below
the leading 1 zero.
This completes the first row. All further operations are carried out on the other rows.

Step 5) Repeat steps 1-4 on the matrix consisting of the remaining rows

The process stops when either no rows remain at Step 5 or the remaining rows consist of
zeros.

43
Q

How to perform Gauss-Jordan algorithm?

A

This brings a matrix to reduced row echelon form

Step 1) Bring matrix to row echelon form using the Gaussian algorithm.

Step 2) Find the row containing the first leading 1 from the right, and add suitable multiples of this row to the rows above it to make each entry above the leading 1 zero.
This completes the first non-zero row from the bottom. All further operations are carried
out on the rows above it.

Step 3) Repeat steps 1-2 on the matrix consisting of the remaining rows.

44
Q

Can all matrices be transformed into row echelon form and reduced row echelon form, and if so, how?

A

(a) Every matrix can be brought to row echelon form by a series of elementary row
operations.

(b) Every matrix can be brought to reduced row echelon form by a series of elementary
row operations.

45
Q

What distinguishes an overdetermined linear system from an underdetermined one?

A

An m×n linear system is:

Overdetermined if it has more equations than unknowns (m>n), which usually (but not necessarily) leads to an inconsistent system.

Underdetermined if it has fewer equations than unknowns (m<n), which means the system may be consistent or inconsistent. If consistent, an underdetermined system typically has infinitely many solutions.

46
Q

What differentiates a homogeneous linear system from an inhomogeneous one?

A

A linear system is considered homogeneous if all of the constant terms bi = 0

If any of the constant terms bi are not zero, the system is inhomogeneous.

For an inhomogeneous system, you can form the associated homogeneous system by setting all the bi’s to zero

47
Q

What can be said about the solution set of an underdetermined homogeneous system?

A

An underdetermined homogeneous system always has non-trivial solutions.

48
Q

When does a square linear system have a unique solution?

A

An n × n system is consistent and has a unique solution, if and only if the only solution of the associated homogeneous system is the zero solution.

49
Q

How is a matrix defined and notated?

A

A matrix is defined as a rectangular array of scalars (real numbers), typically organized in rows and columns. It is notated as
𝐴 = (𝑎𝑖𝑗) 𝑚×𝑛 or simply A =(aij) to denote an mxn matrix, where the entry 𝑎𝑖𝑗 is located in the i-th row and the j-th column.

50
Q

When are matrices considered equality?

A

Two matrices A and B are equal, and we write A = B, if they have the same size and aij = bij where A = (aij ) and B = (bij ).

51
Q

What happens when you multiply a matrix by a scalar?

A

If A = (aij )m×n and α is a scalar, then αA is the m × n matrix whose (i, j)-entry is αaij

52
Q

How is the addition of two matrices defined?

A

If A = (aij )m×n and B = (bij )m×n then the sum A + B of A and B is the m × n matrix whose (i, j)-entry is aij + bij .

53
Q

What is a zero matrix?

A

A zero matrix, or simply O when the size is clear from the context, is an m×n matrix where all entries are zero.

54
Q

Scalar multiplication and addition of matrices satisfy the following rules:

A

Let A, B and C be matrices of the same size, and let α and β be scalars then:

(1) Associative under addition

(2) Commutative under addition

(3) O is the addition identity matrix

(4) -A is the additive inverse

(5) Distributive all ways via scalar multiplication

(6) 1 is a scalar multiplicative identity

55
Q

Matrix Multiplication

A

If A = (aij ) is an m×n matrix and B = (bij ) is an n × p matrix then the product AB of A and B is the m × p matrix C = (cij ) with

cij = sum k=1 to n aikbkj

56
Q

What is an identity matrix?

A

An identity matrix I is a square matrix with 1’s on the diagonal and zeros elsewhere

57
Q

Matrix multiplication rules

A

(a) Matrices are distributive under scalar and Matrix multiplication

(b) The Identity matrix of any size is a multiplicative identity

58
Q

When are two matrices said to commute?

A

If A and B are two matrices with AB = BA, then A and B are said
to commute.

59
Q

What does it mean for a square matrix to have an inverse, and what is an invertible matrix?

A

If A is a square matrix, a matrix B is called an inverse of A if
AB = I and BA = I . A matrix that has an inverse is called invertible.

60
Q

What does it imply if two different matrices are inverses of the same matrix?

A

If B and C are both inverses of A, then B = C.

meaning inverses are unique

61
Q

What is the inverse of a product of two invertible matrices?

A

If A and B are invertible matrices of the same size, then AB is invertible and (AB)−1 = B^−1A^−1
.

62
Q

What is the transpose of a matrix?

A

The transpose of an m × n matrix A = (aij ) is the n × m matrix
B = (bij ) given by bij = aji

63
Q

The main properties of matrix transposition are:

A

(a) (A^T)^T = A. The transpose of a transpose returns the original matrix

(b) (αA^)T = α(A^T). The transpose of a scalar multiple of a matrix is the scalar multiple of the transpose of the matrix

(c) (A + B)^T = A^T + B^T. The transpose of a sum of matrices is the sum of their transposes

(d) (AB)^T = B^TA^T. The transpose of a product of matrices is the product of their transposes in reverse order

64
Q

Inverse of the Transpose of a Matrix

A

Let A be invertible. Then AT
is invertible and

(A^T)^−1 = (A^−1)^T.

65
Q

What characterizes a symmetric matrix?

A

A matrix is said to be symmetric if A^T = A

66
Q

What are the different types of triangular matrices?

A

Upper triangular: if all entries below the main diagonal are zero
if aij = 0 for i > j

Strictly upper triangular: if all entries on and below the main diagonal are zero
if aij = 0 for i ≥ j

Lower triangular: if all entries above the main diagonal are zero
if aij = 0 for i < j

Strictly lower triangular: if all entries on and above the main diagonal are zero
if aij = 0 for i ≤ j

Diagonal: if all off-diagonal entries are zero
if aij = 0 for i ̸= j

67
Q

What is the result of adding or multiplying two upper triangular matrices of the same size?

A

When two upper triangular matrices of the same size are added together or multiplied, the result is also an upper triangular matrix.

68
Q

How does multiplying a linear system by an invertible matrix affect the solution set?

A

Multiplying both sides of a linear system Ax=b by an invertible matrix M does not change the solution set of the system. Thus, the system Ax=b and the modified system MAx=Mb are equivalent, meaning they have the same set of solutions.

69
Q

What is an elementary matrix and how is it created?

A

An elementary matrix of type I (respectively, type II, type III)
is a matrix obtained by applying an elementary row operation of type I (respectively, type
II, type III) to an identity matrix.

70
Q

What is the result of left-multiplying a matrix by an elementary matrix?

A

If E is an m × m elementary matrix obtained from I by an elementary
row operation, then left-multiplying an m × n matrix A by E has the effect of performing
that same row operation on A.

71
Q

What are the properties of the inverse of an elementary matrix?

A

If E is an elementary matrix, then E is invertible and E^−1 is an elementary matrix of the same type.

72
Q

what does it mean for two matrices to be row equivalent?

A

A matrix B is row equivalent to a matrix A if there exists a finite
sequence E1, E2, … , Ek of elementary matrices such that
B = EkEk−1 … E1A

73
Q

Properties of row equivalent matrices

A

(a) A is row equivalent to itself;

(b) if A is row equivalent to B, then B is row equivalent to A;

(c) if A is row equivalent to B, and B is row equivalent to C, then A is row equivalent
to C.

74
Q

What conditions indicate that a square matrix is invertible?

A

(a) The matrix A itself is invertible.

(b) The equation Ax=0 has only the trivial solution, meaning x must be the zero vector.

(c) The matrix A is row equivalent to the identity matrix 𝐼.

(d) The matrix A can be expressed as a product of elementary matrices.

75
Q

What conclusion can be drawn about square matrices A and C if
CA equals the identity matrix I?

A

Then also AC = I. in particular, both A and C are invertible with C = A^−1 and A = C^−1

76
Q

Gauss-Jordan Inversion Method

A

1) Form the augmented matrix (𝐴∣𝐼) with A and the identity matrix I of the same size.

2) Use row reduction to bring (𝐴∣𝐼) to reduced row echelon form.
3) If the left side of the augmented matrix is row equivalent to the identity matrix I, then the right side becomes 𝐴−1, and A is invertible.

4) If the left side cannot be transformed into the identity matrix, then A does not have an inverse.

77
Q

How do you calculate the determinant of a 2x2 matrix?

A

The determinant of A, denoted
det(A), is defined by

det(A) = | a11 a12 |
| a21 a22 |
= = a11a22 - a21a12

78
Q

What are the key properties relating the determinant to the invertibility of 2x2 matrices?

A

(a) det(AB) = det(A) det(B),

(b) det(A) ̸= 0 if and only if A is invertible,

(c)If A =( a c ) is invertible then
( b d )

A^-1 = 1 ( d -c )
det(A) (-b a )

79
Q

How is the determinant of a 3x3 matrix calculated?

A

det(A) = = a11a22a33 − a11a32a23 − a21a12a33 + a21a32a13 + a31a12a23 − a31a22a13

80
Q

How do you define the determinant of an n×n matrix?

A

If n = 1, then det(A) = a11

If n>1, then det(A) is the sum of n terms of the form (-1)^i+1 x ai1det(Ai1) with plus and minus signs alternating. where the entries a11, a21, … , an1 are from the first
column of A

81
Q

What is the cofactor of a matrix and how is it used in calculating the determinant?

A

the (i, j)-cofactor of A is the
number Cij defined by

Cij = (−1)i+j x det(Aij)

Thus, the definition of det(A) reads

det(A) = a11C11 + a21C21 + … + an1Cn1

82
Q

The determinant of an n×n matrix A can be computed using the cofactor expansion across any column or row.What are the equations for both of these

A

The expansion down
the j-th column is

det(A) = a1jC1j + a2jC2j + … + anjCnj

and the cofactor expansion across the i-th row is

det(A) = ai1Ci1 + ai2Ci2 + · · · + ainCin

83
Q

How is the determinant of a triangular matrix determined?

A

If A is either an upper or a lower triangular matrix, then det(A) is the product of the diagonal entries of A.

84
Q

How do different row operations affect the determinant of a square matrix?

A

(a) If two rows of A are interchanged to produce B, then det(B) = − det(A).

(b) If one row of A is multiplied by α to produce B, then det(B) = α det(A).

(c) If a multiple of one row of A is added to another row to produce a matrix B then
det(B) = det(A).

85
Q

When is a matrix invertible?

A

A matrix A is invertible if and only if det(A) ̸= 0.

86
Q

When is a matrix singular?

A

A square matrix A is called singular if det(A) = 0. Otherwise it is said to be nonsingular.

87
Q

Invertibility and Nonsingularity of a Matrix

A

A matrix is invertible if and only if it is nonsingular.

88
Q

How does the determinant of a matrix relate to the determinant of its transpose?

A

If A is an n × n matrix, then
det(A) = det(A^T).

89
Q

How does the application of an elementary matrix affect the determinant of a matrix?

A

If A is an n × n matrix and E an elementary n × n matrix, then
det(EA) = det(E) det(A)

det(E)=−1 if E is of type I (interchanging two rows).

det(E)=α if E is of type II (multiplying a row by α).

det(E)=1 if E is of type III (adding a multiple of one row to another)

90
Q

What is the relationship between the determinants of two square matrices and their product?

A

det(AB) = det(A) det(B)

91
Q

What is the determinant of the inverse of an invertible matrix?

A

If A is an invertible matrix, then the determinant of its inverse, A ^−1, is the reciprocal of the determinant of A.

92
Q

What is Cramer’s rule

A

Det(Bi)/Det(A) = xi where Bi is the matrix A with the ith colum replaced with the coeffecient matrix

Mathematics (6 decks)

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