- What is linear algebra?
- A bad definition then a good one
- Linear algebra is the study of vectors
- What is a vector?
- A vector is not a list of numbers (though a list of numbers can be a vector)
- Children know what vectors are intuitively
- Direction and distance
- No coordinate grid
- Coordinate grids are extra structure and are not mandatory
- Operations on vectors
- Addition
- Scaling
- There are other operations, but they are not fundemental like these.
- Furthermore, anything with an adding and scaling operation, subject to certain properties, can be thought of as a vector
- Examples: Arrows, R^n, polynomials
- We call these vector spaces
- Another bad definition of linear algebra: the study of vector spaces
- What is a vector?
- What is a basis?
- Linear combinations
- A linear combination of a set of vectors is a sum of the vectors, each with a scale factor applied.
- Example
- Linear independence
- A linearly independent set of vectors is one that has no redundant vectors
- In other words, no vector in the set can be written as a sum of the others
- Examples:
- Generating set
- Every vector in the plane can be written as a linear combination of these two vectors:
- Thus these vectors are called a generating (or spanning) set for the plane.
- In general, given a set of vectors you can form the span of them by taking all possible linear combinations of the vectors.
- In this case, the original set is called the generating set of the span
- Basis
- A basis for a vector space is a linearly independent generating set for that space
- In other words, every vector in the space can be written uniquely as a linear combination of the vectors in the basis
- Dimension
- Every basis has the same number of vectors
- This number is called the dimension
- Examples of bases:
- Note that a choice of basis gives a choice of coordinate grid
- Note that choice of basis is entirely arbitrary. Vectors exist independently of bases
- R^3 is a special vector space with a distinguished choice of basis: [1, 0, 0], [0, 1, 0], [0, 0, 1].
- Linear combinations
- What is a linear transform?
- Recall the definition of a function in math:
- a function f takes a vector v and produces an output vector f(v). The vector it produces is always the same given the same input vector
- Functions have a domain and a codomain, representing the vector spaces that the input and output are in.
- If the input space is V and the output space is W we write this as f: V -> W. We read this as "f is a function from V to W".
- A linear transform is a function satisfying an additional property: linearity
- Preservation of addition: f(u+v)=f(u)+f(v)
- Preservation of scaling: f(cv)=cf(v)
- Linearity is sort of like the distributive law and commutative law combined.
- Examples:
- Freshman's dream
- A common error by young or inexperience mathematical discipuli is to assume a nonlinear operation is linear. As an example:
- assuming (x+y)^n=x^n+y^n
- Why is linearity important?
- Lots of reasons: first of all a lot of important functions are linear
- Second of all, they are simpler to deal with, as we shall soon see
- Third, many functions can be approximated by linear functions. Ask if you want further details on this
- Composition of linear functions
- Given any two functions, f and g, you can form the composition of them: f(g(v)).
- This means, first apply the transform g, then apply the transform f to the output of it
- Examples:
- Note that in order for functions to be composable their domains and codomains must match up like f: V -> W and g: U -> V
- Isomorphism
- If every vector in the codomain comes from a unique vector in the domain, we call this an isomorphism
- Examples:
- A choice of a basis for a vector space is the same as choosing an isomorphism to R^n (where n is the dimension).
- We can think of this function as assigning coordinates to every vector in our domain
- A better definition of linear algebra: The study of vector spaces and linear transforms between them
- Recall the definition of a function in math:
- What is a matrix?
- Matrices are to basis as coordinates are to vector spaces
- We note that if the domain of a linear transform is given a choice of basis, then the output of the linear transform is entirely determined by the value of the function on the basis
- Examples:
- From this we can see that in order to represent a basis we can just list out the output vectors of our basis [v_1 v_2 v_3 ... v_n]
- This means that the first vector in our basis goes to v_1, the second vector goes to v_2, etc
- If we give the codomain a basis, then we can write the output vectors in coordinates, and the result of this is the matrix of the transform
- The matrix depends on the choice of basis of both the domain and codomain
- If the domain and codomain are the same vector space, then we can choose a single common basis for both
- Then for each linear transform there is a unique matrix with respect to that basis
- Examples:
- Composition of linear transforms corresponds to matrix multiplication
- I won't go through all the details, but
- If you have two linear transforms, both of which have the same domain and codomain, you may choose a basis and get a matrix for each of them
- You may compose them, getting another linear transform. Hence, the composition has a matrix too
- If you work through the details of this, it turns out that the matrix of the composition corresponds to the matrix multiplication of the matrices of the two transforms
- Example:
- To be continued:
- Further topics
- Change of basis formula
- Gauss-Jordan
- Inner products, orthonormality
- Gram-Schimdt
- Interpolation formula for quaternions
- The first isomorphism theorem for inner product spaces
- Eigenvalues
- Determinants
- Duality
- Tensors
- Further material
- Linear algebra done right by Axler
- Essense of linear algebra by 3blue1brown
- http://number-none.com/product/Understanding%20Slerp,%20Then%20Not%20Using%20It/
- Further topics