Linear algebra for machine learning
- Distance between two points
- Dot product and angle between 2 vectors
- Equation of a line, plane
- Projection of a vector
- Distance of a point from a plane
- Equation of a circle
1. Distance between two points
Let A & B be two points in 2 dimensional geometry.
In case the movement on diagonal side is allowed,
Minkowski distance (Generalisation)
where p = 1, 2, 3 ... if p == 1, minkowski distance will converge into Manhatten distance if p == 2, minkowski distance will converge into Euclidean distance
If A & B points are in n dimensional geometry, then
Euclidean distance(A,B) =
Manhatten distance(A,B) =
Minkowski distance(A,B) =
2. Dot product and angle between 2 vectors
a & b points are in n dimensional vectors, i.e.,
Dot product of a & b = a.b
It is same as matrix multiplication of a & b vectors, i.e.,
Note: By default, a vector is column vector if not mentioned explicitly. So here a & b are column vectors
Also we know that : (proof)
where a = distance of a from origin & b = distance of b from origin
3. Equation of a line, plane
Line in 2D = Plane in 3D = hyper plane in nD
2 dimensional geometry
The equation of a line in 2D is
which can also be written as
If line passes through origin (0, 0) then y-intercept becomes 0. So equation becomes
3 dimensional geometry
Equation of a plane in 3D passing through the origin (0,0)
n dimensional geometry
Equation of a plane in nD passing through the origin (0,0)
We know that if a.b = 0, then a is perpendicular to b ( if = 0, then )
is perpendicular to any point on plane , provided plane passes through origin
4. Projection of a vector
Let & be two vectors and be the angle between them.
is the projection of on
Multiply both sides by
We know that
If is a unit vector, then
5. Distance of a point from a plane
Let be any point in dimensional geometry, which is at a distance from the hyper plane .
Let be a vector passing through origin (0,0)
Distance of point from the plane is the projection of on
projection of on =
If is a unit vector, then
is positive since is less than 90
Similary we can calculate the distance of the point from the plane . Since is greater than , the distance will be negative.
In this way, we can decide in which side of the plane a point lies.
6. Equation of a circle
Let be the center of the circle & be the locus of the center of the circle.
Let be the radius of the circle
If the center of the cirlce is origin i.e., , then the equation of the circle is given as
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