数学英语 周维祺
The study of linear algebraic structures
: the collection of all ordered tuples of real numbers
or
: transpose
: the -th element/item/entry in
A vector/linear space over a field is a set equipped with (vector addition) and (scalar multiplication) satisfying certain properties. Elements from vector spaces are called vectors
Given and , the weighted sum
is called a linear combination of . are called coefficients
If
then are linearly independent otherwise they are linearly dependent
The span of is the set of all their linear combinations:
A maximally linearly independent set
Its cardinality is the dimension of the space, which is independent of the choice of the set
Every element in the space can be uniquely written into a linear combination of basis elements
Plural: bases
The procedure of writing into the linear combination of a given basis (or a given set of vectors)
A function from a vector space is linear if it commutes with and :
A linear function between linear spaces
If both the domain and the range admit finite(or countable) bases, then the linear transformation has a matrix representation.
A system of linear equations
A diagonal matrix is a matrix in which all entries off the main diagonal are
In an upper (resp. lower) triangular matrix, all elements below (resp. above) the main diagonal are
Strictly upper (lower) triangular matrix: an upper (lower) triangular matrix whose main diagonal also vanishes
The range of :
The kernel (null space) of :
Matrix multiplications are compositions of linear transformations
Matrix multiplications are associative but not necessarily commutative
You are given cards, arranged in a matrix form
A red card gives positive points while a black card gives negative points
Each turn you may choose a row or a column and flip its color
Try to make each column sum and each row sum non-negative
Q: Is it always possible to do so regardless of the configuration?
The dimension of the column span (span of column vectors)
If the inverse of exists, then is invertible / non-singular / full rank, otherwise it is singular. The inverse of is
Direct methods (e.g., Gaussian elimination) vs. Iterative methods
A positive definite bilinear (resp. sesquilinear) form
A positive definite sublinear functional that generalizes the notion of length
The Euclidean norm is induced by the Euclidean inner product
A unit vector:
A unimodular vector:
is orthogonal / perpendicular to if
An orthogonal basis is a basis in which elements are mutually orthogonal
An orthonormal basis is an orthogonal basis in which all elements are also unit vectors
If , then is a projection
If in addition, , then is an orthogonal projection
It is important to comprehend mathematics both as a science and as an art.
An exposition of the history helps to build a deeper understanding by seeing how it was developed over time and in various places.
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