Chapter 9: Basic Linear Algebra

Question 1

What is a vector?

Answer 1

A numerical way to describe, record and process spatial information.
A directed line segment. Defined by its length and direction.

Question 2

What is a Matrix?

Answer 2

A numerical way to describe, record and process information about transformations of space

Question 3

Define the sum of vectors and scalar multiple

Answer 3

Sum each component

x.v = x.(v1,v2…) = (x.v1, x.v2, …)

Question 4

What are the properties of vector operations?

Answer 4

Commutative
Associative
vector 0 is the additive unit
vector -v is the additive inverse
Distributive
mixed associativity ab(v) = (ab)v
Multiply with scalar: 1v = v, 0v = vector 0
Multiply with zero vector a0 = vector 0
Multiply with negative scalar/ vector (-a)v = a(-v)

Question 5

Given vectors v1, v2, …vk and scalars a1, a2, …, ak. Give the linear combination with scalar coefficients

Answer 5

a1v1 + a2v2 + … + akvk

Question 6

Give the vector 0 as a linear combination without all coefficients being zero

Answer 6

(1 1 0) - (1 0 1) - (0 1 -1) = (0 0 0)

Question 7

What is a linear transformation?

Answer 7

A function from R^n -> R^m
That preserves vector addition and scalar multiplication in R^n
T(v + w) = Tv + Tw
T(av) = a.T(v)

Question 8

What is a square matrix?

Answer 8

A matrix with the same number of rows and columns

Question 9

What is a zero matrix?

Answer 9

A matrix where all entries are 0

Question 10

When are two matrices equal?

Answer 10

When they have the same number of rows and columns and the corresponding ij components are equal

Question 11

How do we add matrices?

Answer 11

Add corresponding ij components

Question 12

How do we form linear combinations of matrices?

Answer 12

they must all the m x n, perform any multiplication and addition to get one matrix

Question 13

What is the dot product?

Answer 13

The inner product. It is the sum of vkwk for two matrices or vectors

Question 14

How do we multiply matrices?

Answer 14

multiply the elements of the ith row with the jth column in a pairwise fashion and add them up.

A(m x n) x B(n x p) = C(m x p)

Question 15

AB

define pre-multiplied and post-multiplied

Answer 15

B is pre-multiplied by A

A is post-multiplied by B

Question 16

What is the identity matrix?

Answer 16

All entries are 0 apart from those on the main diagonal. In is the multiplicative unit of n x n matrices

Question 17

What is the transpose of matrix A (m x n)?

Answer 17

obtained from A by exchanging the rows with the columns

Question 18

Multiplying an m x n matrix and a n x 1 vector gives us?

Answer 18

an m x 1 vector. We can pull out the vector terms and write it as a linear combination.

Every matrix-vector multiplication gives a linear combination. Conversely, any linear combination has a compact matrix representation.

Question 19

How can a matrix A(m x n) be used to define a linear transformation T?

Answer 19

Multiplying by A defines a function R^n -> R^m
T(v) = Av
A represents linear transformation T

Every linear transformation amounts to pre-multiplication by a matrix

Question 20

What is the standard matrix of T:R^n -> R^m relative to the standard basis vectors (e1, e2, …, en)?

Answer 20

The m x n array of numbers formed from the vectors T(e1), T(e2), …, T(en)
E.g. T(v) = 3v
T(1 0) = 3(1 0) = (3 0)
T(0 1) = 3(0 1) = (0 3)
So the standard matrix of T is (3 0)
(0 3)

Question 21

How do we transpose linear transformations?

Answer 21

T: Rp -> Rm S: Rp -> Rn
(T ° S)v = T(S(v))
and if A and B are the matrix representations of T,S
T(S(v)) = T(Bv) = ABv

Question 22

How do we solve systems of linear equations?

Answer 22

A system of linear equations consists of m equations with n variables. We can arrange the coefficients of the system as a matrix. This is the coefficient matrix. This allows us to write the system of equations in a compact form.

Question 23

What types of solutions can systems of linear equations have?

Answer 23

Unique solution
No Solution - known as inconsistent
Infinite Solutions

Question 24

What are Homogeneous and in-homogeneous systems?

Answer 24

A system Ax = b of m linear equations is called homogeneous when b = 0. Otherwise the system is inhomogeneous.
A homogeneous system Ax = 0 has either one solution or infinitely many solutions. vector x = 0 is always a solution

Question 25

What are the applications of solution sets?

Answer 25

Network Analysis
Linear optimisation
Systems of linear inequalities or polynomials
Linear transformations
Linear combinations

Question 26

What is Gaussian elimination?

Answer 26

The standard method for solving systems of linear equations. Based on the observation that a solution set of a system is not changed by elementary row operations: interchange, multiply, replace with sum of itself and a multiple of another equation.

Question 27

What is row equivalence?

Answer 27

The solution sets for two matrices are the same. Matrix A can be obtained from matrix B

Question 28

Describe the Forward Elimination phase

Answer 28

Apply row operations to transform the given matrix into row-echelon form. (stair case of 1s)

1. find the row with the left most non-zero entry, interchange with first
2. Use row scaling to ensure each entry below the leading entry in the first row is 1
3. Use row addition to ensure each entry below the leading 1 in the first row is 0.
4. Repeat from row 2 down etc.

Question 29

What is row echelon form?

Answer 29

The first non-zero entry in every row is 1
In each row the leading 1 is further to the right than the preceding
Every row of only zeros is below all non-zero rows

Question 30

Describe the Backward Elimination Phase

Answer 30

Solutions are computed from the matrix in row-echelon form. Solutions are computed from the bottom up.

Question 31

What is a parametric solution with a free parameter?

Answer 31

When there are infinitely many solutions