What does R (a) mean in linear algebra It's a capital R. see in the orthogonal section N（A）=R（AT）⊥ What does R () mean I don't think r here means rank. Please look at the question

What does R (a) mean in linear algebra It's a capital R. see in the orthogonal section N（A）=R（AT）⊥ What does R () mean I don't think r here means rank. Please look at the question

Since the range space R (a) = {y | y = ax, where x belongs to V},
The kernel space n (a) = {x | AX = 0, where x belongs to V}
R (at) ⊥, that is R (a ') ⊥. Represents the orthogonal complement space of the range space of the transpose matrix of matrix A
It means that n (a) and R (a ') are orthogonal, so they complement each other

What is linear algebra?

Vector space is an important subject in modern mathematics. As a result, linear algebra is widely used in abstract algebra and functional analysis. Through analytic geometry, linear algebra can be expressed concretely. The theory of linear algebra has been generalized to operator theory. Because the nonlinear model in scientific research can be approximated to linear model, Because of the work of Fermat and Descartes, linear algebra basically appeared in the 17th century. Until the end of the 18th century, the field of linear algebra was only limited to plane and space. In the first half of the 19th century, the transition to n-dimensional vector space was completed. The theory of matrix began in Kelley, and in the second half of the 19th century, the field of linear algebra was limited to plane and space, In 1888, piano defined finite dimensional or infinite dimensional vector space by axiom. Toplitz extended the main theorem of linear algebra to the most general vector space on any field. In most cases, the concept of linear mapping can get rid of matrix calculation and lead to inherent reasoning, In other words, it does not depend on the choice of basis. Instead of commutative body, it uses body or ring which may not be commutative as the domain of definition of operator, which leads to the concept of module. This concept significantly extends the theory of vector space and reorganizes the situation studied in the 19th century, In 1859, Li Shanlan, a famous mathematician and translator in the Qing Dynasty, translated it into "algebra", which is still in use today, Like force, This is the first example of real vector space. Modern linear algebra has been extended to study any or infinite dimensional space. A vector space of dimension n is called an n-dimensional space. In two-dimensional and three-dimensional space, most useful conclusions can be extended to these higher dimensional spaces. Although many people can't easily imagine vectors in n-dimensional space, This kind of vector (i.e. n-tuple) is very effective for representing data. As n-tuple, vector is an "ordered" list of n elements, most people can effectively summarize and manipulate data in this framework. For example, in economics, 8-dimensional vector can be used to represent the gross national product (GNP) of 8 countries, We can use vectors (V1, V2, V3, V4, V5, V6, V7, V8) to show the GNP of each country in a certain year. Here, the GNP of each country is in its own position, Some notable examples are: the group of irreversible linear mappings or matrices, the ring of linear mappings in vector spaces. Linear algebra also plays an important role in mathematical analysis, especially in vector analysis, which describes higher derivatives, tensor products and commutative mappings, A linear operator maps the elements of a linear space to another linear space (or the same linear space) to keep the consistency of addition and scalar multiplication on a vector space. The set of all such transformations is also a vector space. If the basis of a linear space is determined, all linear transformations can be expressed as a number table, It is called matrix. The in-depth study of matrix properties and matrix algorithms (including determinants and eigenvectors) is also considered to be a part of linear algebra. We can simply say that linear problems in Mathematics - those problems showing linearity - are the easiest to be solved. For example, differential calculus studies many linear approximation problems of functions. In practice, it is different from nonlinear problems The method of linear algebra is one of the most important applications in mathematics and engineering. It refers to the method of using the linear point of view to look at the problem, and using the language of linear algebra to describe it and solve it (matrix operation can be used if necessary)

When k satisfies (), vector group A1 = x1-x2 + X3. A2 = x1-x2 + kx3. A3 = kx1 + 2x2 is linearly independent
A: K is not equal to 2, B: K is not equal to 1, C: K is not equal to - 2, D: K is not equal to - 2 and not equal to - 1

Let m1a1 + m2a2 + m3a3 = 0 be linearly independent of x1, X2 and X3, and the equations of M1, M2 and M3 have no non-zero.m1 + M2 + KM3 = 0 (1) - m1-m2 + 2m3 = 0 (2) M1 + km2 = 0 (3) (1) and (2) simultaneously, m3 (K + 2) = 0 (4) must be K + 2 not equal to 0, K not equal to - 2, otherwise the indefinite equations have non-zero solutions, such as (2,1,3 /...)

It is known that XYZ is a real number and satisfies x 2y-5z = - 7, X-Y + Z = 2. Try to compare the relationship between the square of X - the square of Y and the square of Z

x+2y-5z=-7,x-y+z=2,
——》x=z-1,y=2z-3,
——》z^2-(x^2-y^2)=4z^2-10z+8=4(z-5/4)^2+7/4>0,
Namely: x ^ 2-y ^ 2