Linear Algebra Flashcards
Cosine Rule
c^2 = a^2 + b^2 - 2abcosC
What is a vector?
An ordered list of numbers
How to find the magnitude of a vector
How to find the direction of a 2D vector
θ = arctan(u2/u1)
How to add 2 vectors
y + z = (y1 + z1,y2 + z2,…yn + zn)
How to find the distance between 2 vectors
The distance between their tips (magnitude of difference)
How is the dot product defined (both ways)
u.v=(u1v1 +u2v2 +…+unvn)
u.v = |u| |v| cos θ
The dot product is zero
Vectors are orthogonal (perpendicular)
The dot product of a vector with its self
u . u = |u|^2 , so 1 if u is a unit vector
Convert a vector into a unit vector
Divide it by its scalar magnitude
How to project vector v onto vector u
Convert u into a unit vector in both instances (hence the mag squared) and multiply by the dot product
How does a projection (v onto u) work
Stretches “u” such that it is at its minimum distance from “v”. (makes a right angle triangle)
What is a unit vector
A vector with a magnitude of 1
What are standard basis vectors?
The standard unit vectors that can represent any other vector in that dimension, through a linear combination.
How do we represent coordinate spaces
2D -> R^2
3D -> R^3
ND -> R^N
What are the 2 ways of finding the normal to a line in 2D
1) u . v = 0
2) n . p = d
What are the 3 ways of representing vectors as a line in 2D
1) P = P_0 +tu (ND)
2) n . p =d (2D)
3) y = mx + c (2D)
What is the cartesian equation (and formula) for a 2D line
What is the cartesian equation for a 3D line
How are cicles and spheres defined (vector eq)
Why is |p-p0| = r for circles and spheres?
What are the 3 ways in which planes can be defined?
1) p= p_0 + su + tv
2) n . p = d
3) (n_x) x + (n_y) y + (n_z) z = d
What are the 2 things which the cross product can do?
1) Find a othogonal vector to a plane
2) The magnitude is the area of the parallelogram
How to compute cross product
How are circles defined (parametric eq)
p = r cosθ e1 +r sinθ e2
How to find the normal to a circle / sphere
P_1 - P_0 (where p_1 is a point and p_0 is the center)
This is simply the line from the center to the point
How to find a tangent to a circle / sphere at a point
The projection of the point on the tangent to the unit normal = radius. Or without making it unit (p is any point on the line):
How to compute an intersection with a plane
Why does n . p = d work (for a plane)
What is a linear combination?
Vector u is a linear combination over the vector set V if
~~~
u = a_1 v_1 + a_2 v_2 + … + a_n v_n
~~~
Basically a weighted sum of the vectors
What does it mean to for vectors be lineararly (in)dependent?
Dependent: ` a_1 v_1 + … + a_n v_n = 0
Independent: a_1 v_1 + … + a_n v_n /= 0` for all values of a.
Note: can make a dependent set independent by removing a single (or more) vectors
What is a Span of a set of vectors?
Span(V) is the set of all possible vectors generated by the linear combiantion of the vectors in V. Eg: 2 independent vectors span a plane. The span is a vector space
What is a vector subspace
A subset of vectors of another vector space
What is a basis for vector space V?
A linearly independent set of vectors **that span V **through linear combinations. N terms span N dimensions
What are vector coordinates
The scalar values of a linear combination (of a basis):
How to determine vector coordinates
Either through a system of linear equations or by taking the dot product of an orthonormal basis.
What is an orthonormal basis
A basis of vectors which are:
1. Linearly independent
2. Othogonal (perpendicular)
3. Unit vectors
How to create an orthonormal basis using the Gram-Schmidt process?
Works by projecting the vector onto a subspace and taking the difference (which will be orthogonal to the subspace)
How to project a vector onto a subspace (hence finding the closest vector in that space)?
You must have an orthonormal basis.
What is an mxn matrix (aij)
A 2d array of numbers. M rows, N columns. aij referes to row i col j
What is the transpose of a matrix
Swapping values along diagonal (swapping rows vectors with column vectors).
How to add matrices
Scalar multiple of a matrix
What is the Zero matrix 0
All components zero
How does matrix vector multiplication work
Works by scaling each of the columns of the matrix (the stretched versions of the basis) by the respective components of the vectors.
How do matrices relate to the standard basis
Each column of an independent matrix can be defined as where the respective standard basis appears in this new plane of space.
What is the transpose of a vector
How does the dot product relate to a transpose of a vector
In what vector space does the result of a matrix vector multiplication lie
v is domain, v’ is range
What is an identity matrix
Compress a 3D vector into 2D with a matrix
Rotation matrix counter clockwise by theta
Sheer matrix
What is the result of
Multiplication of 2 transposed matrices property
Is matrix multiplication associative and commutative
How is matrix multiplication expressed as the sum of components
What is the geometric meaning of matrix multiplication
One transformation then another BAx = Ax then B
How to create a matrix for a given transformation
Find the transformed standard basis i, j k etc.
What is a linear system
Solution that simultaniously satisfies all equations
What are the three solutions categories for a linear system
- Unique solution
- No solution - inconsistent (eg. parallel)
- Infinite solutions (line, plane etc.)
How to solve a linear system
- Substitution
- Guassian elimination
Matrix notation for linear systems
This means that x can only be found if b is in the column space spanned by A
What is the matrix rank
number of independent columns, or dimensions of column span
Is a solution unique for a linear system with a) dependent columns b) independent columns
a) unique
b) not unique - infinite solutions
What are the 3 operations allowed in gaussian elimination (solving a linear system)
How do you find a solution with Gaussian elimination
Forward elimination into triangular form, then back substitution
How does Gaussian elimination with matrix notation work
In what 2 cases does gaussian elimination breakdown
- Contradiction (eg 1 = 0) -> no solution
- Consistent (eg 0 = 0) -> infinite solution. Solve using x_n = t
What is an inverse matrix
When a matrix is multiplied with its inverse it gives the identity matrix (1s along diagonal, else 0s)
What is the formula for the inverse of a 2x2 matrix
What is the determinant of a 2x2 matrix
|A|= ad - bc
How to compute the determinant of a 3x3+ matrix
|A| = a(ei − fh) − b(di − fg) + c(dh − eg)
What determines the dimension of a linear transformation
The rank of the matrix
What are the 3 requirements for a matrix to have an inverse
- Square nxn
- Determinant not equal to zero
- Independent columns
What is the premise behind the Gauss-Jordan method
A A^(-1) = I
If A^(-1) = [x_1 x_2 ... x_n] (column vectors)
How to compute the Gauss-Jordan method
- Left is A right is I
- Forward elimination into triangular form
- Back elimination into diagonal
- Scalar multiple so left becomes identity matrix
- Right matrix is now A^(-1)
What is cramer’s rule for solving linear systems
Replace ith column with b (Av = b) and solve.
Concept of determinant representing area (or coordinates)
Determinant of a nxn matrix
What is an eigen vector?
What is an eigen value?
How many eigen vectors exist for each eigen value?
How many eigen values exist for a NxN matrix
How to find the eigen values of a matrix
How to find the eigen vectors of a matrix
How to raise a matrix to a power using diagonalisation
What are markov chains
What are transition matrices
What is the simplified difference equation and the steady state
What is the convergence rate dependent upon
What are the eigen vectors and eigen values of the inverse of A
Same eigen vectors
1/eigen values
