Chapter 9: Basic Linear Algebra Flashcards
(31 cards)
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
