Let u be all n * n upper triangular matrices and l be n * n lower triangular matrices. How to prove that u ⊕ L = R ^ n * n? R ^ n * n is defined as u ∪ l-u ∩ L for all n * n matrices

According to your definition, it's all half angle matrices except diagonal matrices, which obviously can't be R ^ n * n
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Ask about the definition of upper triangular matrix and whether zero matrix is upper triangular matrix

First of all, it must be a square matrix, that is, the rows and columns are equal. The upper triangular matrix is a matrix whose elements below the diagonal are all 0. Of course, the zero matrix is also an upper triangular matrix

What does triangle mean in mathematics?
Hurry!

There are three groups of operation relations among trigonometric functions of the same angle
Trigonometric functions are periodic functions with 2 π as the period
The basic identity of trigonometric function has sum angle formula
sin(a＋β)＝sinαcosβ＋cosαsinβ
cos(a＋β)＝cosαcosβ－sinαsinβ
From these two formulas, we can derive the difference angle formula, double angle formula, half angle formula, sum difference product and integration sum difference formula
The solution of a triangle is to find the other unknown elements when some elements (edges and angles) of a triangle are known. Suppose that the three angles of a triangle are a, B and C, and the opposite sides are a, B and C, respectively
Sine theorem: A / Sina = B / SINB = C / sinc = 2R
Cosine theorem: a 2 = B 2 + C 2-2bccos a these two theorems are the main basis for solving triangles
Trigonometric equation generally refers to the equation containing some trigonometric functions, and the independent variables of trigonometric functions contain unknowns. Since every trigonometric function is a periodic function, any trigonometric equation has infinitely many solutions as long as it has solutions
Triangulation
Triangulation refers to the technology of accurately measuring distance and angle in navigation, surveying and civil engineering. It is mainly used for positioning ships or aircraft. Its principle is that if one side and two corners of a triangle are known, the other two corners can be calculated by plane trigonometry, Pythagoras, a famous mathematician in ancient Greece, proved for the first time the Pythagorean theorem about right triangle, that is, the Pythagorean theorem in China, which made a great contribution to the study and application of geometry

Mathematics - matrix - what does a t mean in the upper right corner of a matrix?
As shown in the picture
What does a t mean in the upper right corner of a matrix?
How to calculate?

For matrix transpose, it can also be represented by a ', for example:
1 2
5 6
A=-2 0
7 9;
A'=1 5 -2 7
2 6 0 9
Row column interchange

Prove the product of two upper triangular matrices of order n
Third order experiment, but the result is wrong,
such as
111 111 321
110* 110= 221
100 100 111

You've got the definition of upper triangular matrix wrong,
----------All elements below the main diagonal are zero

It is proved that the sum, difference, multiplication and product of upper triangular matrix are still triangular matrix

This is no special method, very simple, as long as you set up two upper triangular matrix, according to the operation, calculate the result, determine is still the upper triangular matrix. Not difficult, write it yourself

Prove that the product of upper triangular matrix is still upper triangular matrix!

Let a = (AIJ), B = (bij) be an upper triangular matrix of order n, then AIJ = bij = 0 when I > J. let C = AB = (CIJ) then CIJ = ai1b1j +... + aii-1bi-1j + AI, IBI

Let a be an upper triangular matrix of order n, and the element on the main diagonal is not zero. How to prove that its inverse matrix is also an upper triangular matrix?

Proof: using the method of adjoint matrix
It is reversible by a, a ^ - 1 = a * / | a|
A = (AIJ), a * = (AIJ) ^ t
Where AIJ = (- 1) ^ mij is AIJ's algebraic cofactor, mij is AIJ's cofactor
When II
2. A line multiplied by a nonzero constant
In these two kinds of transformations, the right block always keeps the form of upper triangle
So the final a ^ - 1 is an upper triangular matrix

It is proved that if a is an upper triangular matrix with all principal diagonal elements zero, then a ^ 2 is also an upper triangular matrix with all principal diagonal elements zero

Proof of definition

Can principal diagonal elements of upper triangular matrix be all zero?

Yes! As long as the lower right corner is zero!

Let u be all n * n upper triangular matrices and l be n * n lower triangular matrices. How to prove that u ⊕ L = R ^ n * n? R ^ n * n is defined as u ∪ l-u ∩ L for all n * n matrices