The high number of linear dependence, linear independence, the inner product of the dot product, the concept of projection

1. The linear correlation, linear independence

Linear algebra, a group of elements in the vector space, if there is no available vector finite linear combination of other vectors indicated, is called linearly independent or linearly independent  (linearly independent), otherwise known as linear correlation (linearly dependent).

definition:

In the vector space V of a set of vectors A:   , if there is not all zeros the number k1, k2, ···, km, so that

A group called vectors are linearly related  , otherwise the number of k1, k2, ···, km-wide is 0, it is called linearly independent.

Thereby defining see   whether linear correlation, to see whether there is an incomplete set of numbers zero k1, k2, ···, km such that the equation holds.

That is to look at  the homogeneous linear equations whether there is a non-zero solution, which is the simplest form the coefficient matrix into a matrix, can be solved.

Further, when the coefficient matrix is homogeneous linear equations is a square matrix, the coefficient matrix determinant is present 0, i.e., non-zero solution, whereby  linear correlation.

 Such as: 

There are three numbers a, b, c
If there is a failure of the three numbers m 0, n, k
such that ma + nb + kc = 0
said a, b, c or linear correlation if only when m = n = k = 0 when established, they are linearly independent
in fact a, thing b, c on behalf of the many, not necessarily the number, also can be vectors ah, and so
the number is not necessarily the three, this is just an example, it can be unlimited more

 

 2. within the plot, dot (dot product)

In mathematics, the dot product (dot product; scalar product, also known as the dot product) is accepted in the two vectors R and returns a real number a real-valued scalar binary operation. It is the standard Euclidean space of the product.

Two vectors a = [a1, a2, ..., an] and b = [b1, b2, ..., bn] The dot product is defined as:

a·b=a1b1+a2b2+……+anbn。

And using the matrix multiplication (column) vectors as n × 1 matrix, the dot product may also be written as:

a · b = a * b ^ T, where a ^ T indicates a matrix A transposing.

 

Broadly defined

In a vector space V, defined   positive definite symmetric bilinear form that is a function of the number of volume V, there is added a scalar product of the vector space that is an inner product space.

Algebraic Definition

Provided the two-dimensional vector space   and   define their scalar product (also called inner product, dot product) of the real numbers:

 

 

More generally, the n-dimensional vector inner product is defined as follows:

 

Geometric definition

The two-dimensional space is provided with two vectors   and   ,   and the   vector representing the size b and a, which angle is   , the inner product is defined as a real number:

 This definition is valid only for two-dimensional and three-dimensional space.

This operation can be simply understood as:

In the dot product of vectors, the first vector onto a second (here, the sequence of the vector is not important, the dot product is exchangeable), then "normalized" by dividing by the length of their scalar.

Thus, this score must be less, it may simply be converted to an angle value of one.

 

 3. Projection

Linear algebra and functional analysis, the projection is a mapping from a vector space to a linear transformation itself, and in daily life formal generalized the concept of "parallel projection."

In reality things with sunlight projected onto the ground like projection transformation will map the entire vector space to one of its sub-space, and is identical transformation in this subspace.

If the vector inner product space is given, it can be defined orthogonal and other related concepts (such as a linear operator of the self-adjoint) a.

Inner product space (vector space gives the inner product), the concept of orthogonal projection. Specifically, it refers to an orthogonal projection image space U and W null space of mutually orthogonal subspace projection.

 definition:  

 Projection strict definition is: V strikes a linear transformation from its own vector space P is a projection, if and only if.

Further definition is more intuitive: P is a projection, if and only if there is a subspace W V, such P all the elements V and W are mapped to, and P is an identity transform on W.

Mathematical language to describe, is this:  that   , and   .  

 

 Quoted from the article: "Baidu Encyclopedia"