Chapter 9: Basic Linear Algebra Flashcards
What is a vector?
A numerical way to describe, record and process spatial information.
A directed line segment. Defined by its length and direction.
What is a Matrix?
A numerical way to describe, record and process information about transformations of space
Define the sum of vectors and scalar multiple
Sum each component
x.v = x.(v1,v2…) = (x.v1, x.v2, …)
What are the properties of vector operations?
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)
Given vectors v1, v2, …vk and scalars a1, a2, …, ak. Give the linear combination with scalar coefficients
a1v1 + a2v2 + … + akvk
Give the vector 0 as a linear combination without all coefficients being zero
(1 1 0) - (1 0 1) - (0 1 -1) = (0 0 0)
What is a linear transformation?
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)
What is a square matrix?
A matrix with the same number of rows and columns
What is a zero matrix?
A matrix where all entries are 0
When are two matrices equal?
When they have the same number of rows and columns and the corresponding ij components are equal
How do we add matrices?
Add corresponding ij components
How do we form linear combinations of matrices?
they must all the m x n, perform any multiplication and addition to get one matrix
What is the dot product?
The inner product. It is the sum of vkwk for two matrices or vectors
How do we multiply matrices?
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)
AB
define pre-multiplied and post-multiplied
B is pre-multiplied by A
A is post-multiplied by B
What is the identity matrix?
All entries are 0 apart from those on the main diagonal. In is the multiplicative unit of n x n matrices
What is the transpose of matrix A (m x n)?
obtained from A by exchanging the rows with the columns
Multiplying an m x n matrix and a n x 1 vector gives us?
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.
How can a matrix A(m x n) be used to define a linear transformation T?
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
What is the standard matrix of T:R^n -> R^m relative to the standard basis vectors (e1, e2, …, en)?
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)
How do we transpose linear transformations?
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
How do we solve systems of linear equations?
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.
What types of solutions can systems of linear equations have?
Unique solution
No Solution - known as inconsistent
Infinite Solutions
What are Homogeneous and in-homogeneous systems?
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
What are the applications of solution sets?
Network Analysis Linear optimisation Systems of linear inequalities or polynomials Linear transformations Linear combinations
What is Gaussian elimination?
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.
What is row equivalence?
The solution sets for two matrices are the same. Matrix A can be obtained from matrix B
Describe the Forward Elimination phase
Apply row operations to transform the given matrix into row-echelon form. (stair case of 1s)
- find the row with the left most non-zero entry, interchange with first
- Use row scaling to ensure each entry below the leading entry in the first row is 1
- Use row addition to ensure each entry below the leading 1 in the first row is 0.
- Repeat from row 2 down etc.
What is row echelon form?
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
Describe the Backward Elimination Phase
Solutions are computed from the matrix in row-echelon form. Solutions are computed from the bottom up.
What is a parametric solution with a free parameter?
When there are infinitely many solutions
