Vectors and Matrices

Question 1

What is the vector AB

Answer 1

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

Question 2

What is the defenition of a position vector OA

Answer 2

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

Question 3

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

Answer 3

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)

Question 4

What is the sum of the vectors OA + OB

Answer 4

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

Question 5

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

Answer 5

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

Question 6

Addition and Multiplication properties in R

Answer 6

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

Question 7

Addition and Scalar Multiplication in R^n

Answer 7

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

Question 8

What is a Vector space?

Answer 8

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

Question 9

What is the standard vector ei.

Answer 9

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.

Question 10

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

Answer 10

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

Question 11

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

Answer 11

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

Question 12

What is the Unit vector?

Answer 12

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

Question 13

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.

Answer 13

|v| is the the length of the segment OP

Question 14

What is the vector equation for a line l?

Answer 14

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

Question 15

What are the parametric equations of a line l

Answer 15

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

Question 16

What is the cartesian equation of a line l

Answer 16

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

Question 17

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

Answer 17

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

Question 18

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?

Answer 18

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.

Question 19

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

Answer 19

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

Question 20

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

Answer 20

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.

Question 21

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

Answer 21

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.

Question 22

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

Answer 22

1) Commutative

2) Distributive over vector addition

3) Distributive over vector addition (other order)

4) Compatible with scalar multiplication

Question 23

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

Answer 23

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

|u · v| ≤ |u||v|.

Question 24

What is the Triangle inequality?

Answer 24

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

|u + v| ≤ |u| + |v|.

Question 25

What is the vector equation of a plane?

Answer 25

r · n = p · n

Question 26

What is the cartesian equation for a plane?

Answer 26

ax + by + cz = d where d = n1p1 + n2p2 + n3p3

Question 27

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

Answer 27

The distance D between point Q and plane Π is given by D = | q⋅n−d | |n|

Question 28

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

Answer 28

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

Question 29

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

Answer 29

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

Question 30

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

Answer 30

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 | = |u||v|sin θ if u ̸= 0, v ̸= 0 |u × v| is equal to the area of the parallelogram

Question 31

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

Answer 31

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

Question 32

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

Answer 32

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|

Question 33

What is an inconsistent system?

Answer 33

A system with no solution is called inconsistent

Question 34

What is a consistent system?

Answer 34

A system with at least one solution is called consistent

Question 35

What is the solution set?

Answer 35

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

Question 36

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

Answer 36

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

Question 37

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

Answer 37

(i) interchanging two equations; (ii) multiplying an equation by a non-zero scalar; (iii) adding a multiple of one equation to another.

Question 38

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

Answer 38

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.

Question 39

What are the elementary row operation?

Answer 39

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.

Question 40

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

Answer 40

(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.

Question 41

What are leading and free variables?

Answer 41

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.

Question 42

How to perform Gaussian algorithm

Answer 42

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.

Question 43

How to perform Gauss-Jordan algorithm?

Answer 43

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.

Question 44

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

Answer 44

(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.

Question 45

What distinguishes an overdetermined linear system from an underdetermined one?

Answer 45

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

Question 46

What differentiates a homogeneous linear system from an inhomogeneous one?

Answer 46

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

Question 47

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

Answer 47

An underdetermined homogeneous system always has non-trivial solutions.

Question 48

When does a square linear system have a unique solution?

Answer 48

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.

Question 49

How is a matrix defined and notated?

Answer 49

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.

Question 50

When are matrices considered equality?

Answer 50

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 ).

Question 51

What happens when you multiply a matrix by a scalar?

Answer 51

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

Question 52

How is the addition of two matrices defined?

Answer 52

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 .

Question 53

What is a zero matrix?

Answer 53

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

Question 54

Scalar multiplication and addition of matrices satisfy the following rules:

Answer 54

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

Question 55

Matrix Multiplication

Answer 55

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

Question 56

What is an identity matrix?

Answer 56

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

Question 57

Matrix multiplication rules

Answer 57

(a) Matrices are distributive under scalar and Matrix multiplication (b) The Identity matrix of any size is a multiplicative identity

Question 58

When are two matrices said to commute?

Answer 58

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

Question 59

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

Answer 59

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.

Question 60

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

Answer 60

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

Question 61

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

Answer 61

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

Question 62

What is the transpose of a matrix?

Answer 62

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

Question 63

The main properties of matrix transposition are:

Answer 63

(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

Question 64

Inverse of the Transpose of a Matrix

Answer 64

Let A be invertible. Then AT is invertible and (A^T)^−1 = (A^−1)^T.

Question 65

What characterizes a symmetric matrix?

Answer 65

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

Question 66

What are the different types of triangular matrices?

Answer 66

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

Question 67

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

Answer 67

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

Question 68

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

Answer 68

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.

Question 69

What is an elementary matrix and how is it created?

Answer 69

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.

Question 70

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

Answer 70

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.

Question 71

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

Answer 71

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

Question 72

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

Answer 72

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

Question 73

Properties of row equivalent matrices

Answer 73

(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.

Question 74

What conditions indicate that a square matrix is invertible?

Answer 74

(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.

Question 75

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

Answer 75

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

Question 76

Gauss-Jordan Inversion Method

Answer 76

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.

Question 77

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

Answer 77

The determinant of A, denoted det(A), is defined by det(A) = | a11 a12 | | a21 a22 | = = a11a22 - a21a12

Question 78

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

Answer 78

(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 )

Question 79

How is the determinant of a 3x3 matrix calculated?

Answer 79

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

Question 80

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

Answer 80

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

Question 81

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

Answer 81

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

Question 82

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

Answer 82

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

Question 83

How is the determinant of a triangular matrix determined?

Answer 83

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

Question 84

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

Answer 84

(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).

Question 85

When is a matrix invertible?

Answer 85

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

Question 86

When is a matrix singular?

Answer 86

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

Question 87

Invertibility and Nonsingularity of a Matrix

Answer 87

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

Question 88

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

Answer 88

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

Question 89

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

Answer 89

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)

Question 90

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

Answer 90

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

Question 91

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

Answer 91

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

Question 92

What is Cramer's rule

Answer 92

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