Linear Algebra 1
Matrices, Matrix multiplication ,Inverse Matrix, Matrices to solve a system of equations, Matrices to solve a vector combination problem ,Singular Matrices , 3-variable linear equations ,Introduction to Vectors, Parametric Representations of Lines, Linear Combinations and Span, Linear Independence ,Linear Subspaces
Product and Vector Length, Proving Vector Dot Product Properties ,Proof of the
Cauchy-Schwarz Inequality, Proof of the Cauchy-Schwarz Inequality; Proof of the Cauchy-Schwarz Inequality,
Vector Triangle Inequality, Vector Triangle Inequality; Proving the triangle inequality for vectors
in Rn, Defining the angle between vectors, Defining the angle between
vectors; Introducing the idea of an angle
between two vectors, Defining a plane in R3 with a point and normal vector,
Cross Product Introduction, Relationship between cross product and sin of
angle, Dot and Cross Product Comparison/Intuition, Dot and Cross Product
Comparison/Intuition; Dot and Cross
Product Comparison/Intuition, matrices: Reduced Row Echelon Form , Matrix
Vector Products, Null Space of a Matrix ,Calculating the null space of a
matrix, Column Space of a Matrix ,Visualizing a Column Space as a Plane in
R3,Proof: Any subspace basis has same number of elements, Dimension of the Null
Space or Nullity , Dimension of the Column Space or Rank ,relation between
basis cols and pivot cols; ,Showing that
the candidate basis does span C(A)
Linear Algebra 3
Transformations, Linear Transformations, Matrix Vector Products as Linear
Transformations, Linear Transformations as Matrix Vector Products, Image of a
subset under a transformation, im (T): Image of a Transformation, Preimage of a
set, Preimage and Kernel Example, Sums and Scalar Multiples of Linear
Transformations, Rotation in R3 around the X-axis, Unit Vectors, Matrix Product
Examples, Matrix Product Associativity, Distributive Property of Matrix
A line passing through the origin (blue, thick) in R3
is a linear subspace, a common object of study in linear algebra.
Linear algebra is a branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations),
and systems of linear equations. Vector spaces are a central theme in
modern mathematics; thus, linear algebra is widely used in both
abstract algebra and functional analysis. Linear algebra also has a
concrete representation in analytic geometry and it is generalized in
operator theory. It has extensive applications in the natural sciences
and the social sciences, since nonlinear models can often be
approximated by linear ones.
had its beginnings in the study of vectors in Cartesian 2-space and
3-space. A vector, here, is a directed line segment,
characterized by both its magnitude (also called length or norm) and
its direction. The zero vector is an exception; it has zero magnitude
and no direction. Vectors can be used to represent physical entities
such as forces, and they can be added to each other and multiplied by
scalars, thus forming the first example of a real vector space, where a
distinction is made between "scalars", in this case real numbers, and
Modern linear algebra has been extended to consider spaces of arbitrary or infinite dimension. A vector space of dimension n is called an n-space.
Most of the useful results from 2- and 3-space can be extended to these
higher dimensional spaces. Although people cannot easily visualize
vectors in n-space, such vectors or n-tuples are useful in representing data. Since vectors, as n-tuples,
consist of n ordered components, data can be efficiently summarized and
manipulated in this framework. For example, in economics, one can
create and use, say, 8-dimensional vectors or 8-tuples to represent the
gross national product
of 8 countries. One can decide to display the GNP of 8 countries for a
particular year, where the countries' order is specified, for example,
(United States, United Kingdom, France, Germany, Italy, Japan,
Switzerland and Belgium), by using a vector (v1, v2, v3, v4, v5, v6, v7, v8) where each country's GNP is in its respective position.
A vector space (or linear space), as a purely abstract concept about
which theorems are proved, is part of abstract algebra, and is well
integrated into this discipline. Some striking examples of this are the
group of invertible linear maps or matrices, and the ring of linear
maps of a vector space.
Linear algebra also plays an important part in analysis, notably, in
the description of higher order derivatives in vector analysis and the
study of tensor products and alternating maps.
In this abstract setting, the scalars with which an element of a
vector space can be multiplied need not be numbers. The only
requirement is that the scalars form a mathematical structure, called a
field. In applications, this field is usually the field of real numbers
or the field of complex numbers. Linear maps
take elements from a linear space to another (or to itself), in a
manner that is compatible with the addition and scalar multiplication
given on the vector space(s). The set of all such transformations is
itself a vector space. If a basis for a vector space is fixed, every
linear transformation can be represented by a table of numbers called a
matrix. The detailed study of the properties of and algorithms acting
on matrices, including determinants and eigenvectors, is considered to
be part of linear algebra.
can say quite simply that the linear problems of mathematics—those that
exhibit linearity in their behavior—are those most likely to be solved.
For example, differential calculus does a great deal with linear
approximation to functions. The difference from nonlinear problems is
very important in practice.
- Every vector space has a basis.
- Any two bases of the same vector space have the same cardinality; equivalently, the dimension of a vector space is well-defined.
- A square matrix is invertible if and only if its determinant is nonzero.
- A matrix is invertible if and only if the linear map represented by the matrix is an isomorphism.
a square matrix has a left inverse or a right inverse then it is
invertible (see invertible matrix for other equivalent statements).
- A matrix is positive semidefinite if and only if each of its eigenvalues is greater than or equal to zero.
- A matrix is positive definite if and only if each of its eigenvalues is greater than zero.
- An n×n matrix is diagonalizable (i.e. there exists an invertible matrix P and a diagonal matrix D such that A = PDP−1) if and only if it has n linearly independent eigenvectors.
- The spectral theorem states that a matrix is orthogonally diagonalizable if and only if it is symmetric.
Specific entries of a matrix are often referenced by using pairs of subscripts.
A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers, such as
An item in a matrix is called an entry or an element. The
has entries 1, 9, 13, 20, 55, and 4. Entries are often denoted by a
variable with two subscripts, as shown on the right. Matrices of the
same size can be added and subtracted entrywise and matrices of
compatible size can be multiplied. These operations have many of the
properties of ordinary arithmetic, except that matrix multiplication is
not commutative, that is, AB
are not equal in general. Matrices consisting of only one column or row
define the components of vectors,
while higher-dimensional, e.g. three-dimensional, arrays of numbers
define the components of a generalization of a vector called a tensor.
Matrices with entries in other fields or rings are also studied.
Matrices are a key tool in linear algebra. One use of matrices is to
represent linear transformations, which are higher-dimensional analogs
of linear functions of the form f(x) = cx, where c
is a constant; matrix multiplication corresponds to composition of
linear transformations. Matrices can also keep track of the
coefficients in a system of linear equations. For a square matrix, the
determinant and inverse matrix (when it exists) govern the behavior of
solutions to the corresponding system of linear equations, and
eigenvalues and eigenvectors provide insight into the geometry of the
associated linear transformation.
Matrices find many applications. Physics makes use of matrices in
various domains, for example in geometrical optics and matrix
mechanics; the latter led to studying in more detail matrices with an
infinite number of rows and columns. Graph theory uses matrices to keep
track of distances between pairs of vertices in a graph. Computer
graphics uses matrices to project 3-dimensional space onto a
2-dimensional screen. Matrix calculus generalizes classical analytical
notions such as derivatives of functions or exponentials to matrices.
The latter is a recurring need in solving ordinary differential
equations. Serialism and dodecaphonism are musical movements of the
20th century that use a square mathematical matrix to determine the
pattern of music intervals.
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