Non-degraded bilinear linear type

Non-degraded bilinear is a very useful concept. It can map the problem "image" in a space to its dual space and solve it in this dual space. This section briefly introduces the basic knowledge of non-degraded bilinear series. This article mainly refers to the book "linear algebra: An Introductory approach" in Curtis, which is a very down-to-earth book. It is very suitable for beginners to learn more. The description here should be concise. If "obviously", "not difficult to verify" and other terms appear in the text, it means that the reasoning in this part is really simple and will not be detailed again. However, I would like to remind you that, although the author is so accustomed to the nature of these basics that he is too lazy to come up with verification, for those who read this part for the first time, it is recommended that you verify it one by one. This is part of the learning process that cannot be lazy.

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What is a non-degraded bilinear type?

Set $ V, W $ to the finite dimension vector space on the domain $ F $, ing $ B (V, W ): V \ times w \ rightarrow F $ is called a bilinear function, if it is linear for each component:

\ [\ Begin {Align *} B (a_1v_1 + a_2v_2, W) & = a_1b (v_1, W) + a_2b (V_2, W), \ B (V, a_1w_1 + a_2w_2) & = a_1b (v, W_1) + a_2b (v, W_2 ). \ end {Align *} \]

If bilinear functions $ B $ meet the following conditions:

1. If $ V \ In V $ makes $ w \ in W $ B (V, W) = 0 $, $ V = 0 $.

2. If $ w \ in W $ causes $ B (V, W) = 0 $ for any $ V \ in W $, then $ W = 0 $.

$ B $ is called a non-degraded bilinear function.

Assume that $ B (V, W): V \ times w \ rightarrow F $ is a non-degraded bilinear linear type. For the given $ V \ In V $, you can define a linear function $ \ varphi_v on $ W $:
\ [\ Varphi_v (w) = B (V, W). \]
Ing $ V \ rightarrow \ varphi_v $ is a ing between $ V $ and $ W $ dual space $ w ^ \ ast $. This ing is obviously linear, in addition, it is easy to verify that it is a single shot, and thus $ \ dim V \ Leq \ dim w ^ \ ast $. Likewise, linear ing
\ [W \ rightarrow \ phi_w: \ phi_w (v) = B (V, W) \]
Is a single shot from $ W $ to $ V ^ \ ast $, so $ \ dim w \ Leq \ dim V ^ \ ast $. Because any finite dimension vector space has the same dimension as its dual space, we obtain $ \ dim v = \ dim W $ by combining the two inequalities, and $ V \ rightarrow \ varphi_v $ is a linear homogeneous structure from $ V $ to $ w ^ \ ast $, $ w \ rightarrow \ phi_w $ is a linear homogeneous structure from $ W $ to $ V ^ \ ast $.

To sum up, if $ V \ times w \ rightarrow F $ is a non-degraded bilinear type $ B $, then $ \ dim v = \ dim W $, in addition, $ V $ and $ W $ can be mapped to the dual space of the other party by means of $ B $. In this case, $ V $ and $ W $ are mutually dual spaces.

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Zero subaccount

Set $ v_1 \ subset v $ to the sub-space of $ V $, and define its subdividing sub about $ B $ v_1 ^ \ bot \ subset W $
\ [V_1 ^ {\ bot }=\{ w \ In w \ | \ B (V, W) = 0, \ forall V \ In v_1 \}. \] Similarly, for the sub-space $ W_1 $ of $ W $, it can be defined as a zero sub. $ W_1 ^ \ bot \ subset v $ is
\ [W_1 ^ {\ bot }=\{ V \ In V \ | \ B (V, W) = 0, \ forall w \ In W_1 \}. \] because of the dual-line nature of $ B $, it is not difficult to verify that $ v_1 ^ \ BOT and W_1 ^ \ bot $ are all sub-spaces.

Defines bilinear functions $ B _1: v_1 \ times w/v_1 ^ \ bot \ rightarrow F $
\ [B _1 (v_1, [w]) = B (v_1, W), \ quad v_1 \ In v_1, w \ in W. \]
It is not difficult to verify that the definition of $ B _1 $ is reasonable (not dependent on the selection of representative elements in the commercial space) and non-degraded, so that we can get
\ [\ Dim v_1 = \ dim w/v_1 ^ \ bot = \ dim w-\ dim v_1 ^ \ bot. \]

The following is a very useful conclusion: If $ v = v_1 \ oplus V_2 $ is the forward sum of the sub-spaces, $ W $ also breaks down the straight sum of the corresponding zero Sub: $ W = v_1 ^ \ bot \ oplus V_2 ^ \ bot $ (proving to be ordinary ).

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Adjoint transformation

Definition:Set $ A: V \ rightarrow v $ to a linear transformation, and define the adjoint transformation of $ A $ (about bilinear type $ B $) $ A ^ \ AST: w \ rightarrow W $ is the only linear transformation that satisfies the following equations: \ [B (AV, W) = B (V, A ^ \ ast W) \ quad V \ In V, W \ in W. \]

The definition of this sentence is refreshing, but the problem is that $ A ^ \ ast $ meets the requirements? Is it unique if it exists? Here we will solve these two problems.

Set $ \ Text {end} (w) $ to the space composed of all linear transformations on $ W $, $ \ Text {Bil} (V, W) $ is the vector space composed of all bilinear functions on $ V \ times W $.

For any $ f \ In \ Text {end} (w) $, define the bilinear function on $ V \ times W $ B _f $ as $ B _f = B (V, FW) $. Ing $ f \ rightarrow B _f $ is obviously linear, and it is a single shot with the non-fading ability of $ B $, therefore, $ f \ rightarrow B _f $ is a linear single shot from $ \ Text {end} (w) $ to $ \ Text {Bil} (V, W) $. However, the dimensions of $ \ Text {end} (w) $ and $ \ Text {Bil} (V, W) $ are the same. They are both $ \ dim V \ times \ dim W $, this is a linear homogeneous structure. Therefore, any element in $ \ Text {Bil} (V, W) $ is like $ B (v, Fw) $, $ f \ In \ Text {end} (w) $ is uniquely identified. In particular, we apply this conclusion to the bilinear type $ B (AV, W) $, and we can see that there is a unique $ A ^ \ ast \ In \ Text {end} (W) $ makes $ B (AV, W) = B (V, A ^ \ ast W) $ true, which proves the conclusion.

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Limit vs Induction

Assume that $ A $ is a linear transformation on $ V $, and the sub-space $ v_1 $ is the constant sub-space of $ A $, it is easy to verify that $ v_1 ^ \ bot $ is the constant sub-space of $ A ^ \ ast $, therefore, we can consider the limitations of $ A $ on $ v_1 $ A_1 = A | _ {v_1} $, and $ A ^ \ ast $ induction on the commercial space $ w/v_1 ^ \ bot $ \ widetilde {A ^ \ ast} $.

Theorem:$ A_1 $ and $ \ widetilde {A ^ \ ast} $ about bilinear type $ B _1 (v_1, [w]) = B (v_1, W) $ mutual adjoint transformation:

\ [B _1 (A_1 v_1, [w]) = B _1 (v_1, \ widetilde {A ^ \ ast} [w]), \ quad v_1 \ In v_1, [w] \ in W/v_1 ^ \ bot. \]

This is a very important conclusion. It is also an important and general conclusion in modern mathematics.

Proof: according to the definition of the induced transformation, $ \ widetilde {A ^ \ ast} [w] = [A ^ \ ast W] $. Therefore

\ [B _1 (v_1, \ widetilde {A ^ \ ast} [w]) = B _1 (v_1, [a ^ \ ast W]) = B (v_1, A ^ \ ast W) = B (av_1, W) = B _1 (a_1v_1, [w]). \]

So far, let's make a conclusion: In the non-degraded bilinear theory that has been introduced, there is a high degree of symmetry everywhere: $ V $ and $ W $ are mutually dual spaces. The zero sub-sets up a one-to-one correspondence between the $ V $ sub-spaces and the $ W $ sub-spaces (even the sub-spaces are directly and decomposed; the linear transformation on $ V $ corresponds to the linear transformation on $ W $. The limitation corresponds to the induction. This high symmetry is very suitable for transferring the problem in $ V $ to the dual space $. The following is a wonderful application of this idea.

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Concise Proof of Rational Standard Form

This section provides a very concise proof of Rational Standard Form theorem. This process can be seen as a "typical usage" of non-degraded bilinear series ".

For each vector $ V \ In V $, define its order as $ m_v (x) \ In F [x] $. Here $ m_v (X) $ is the first polynomial that minimizes the number of times $ m_v (a) V = 0 $. Any polynomial $ g (x) $ that makes $ G (a) V = 0 $ is a double of $ m_v (x) $.

Theorem:Set $ A $ to linear transformation on the finite dimension vector space on the domain $ F $, a set of non-zero vectors exist in $ V $ \ {v_1, \ ldots, v_r \}$. Their order is the power of the prime polynomial $ \ {P_1 (x) ^ {E_1}, \ ldots, p_r (x) ^ {e_r }\}$, and $ V $ is the cyclic subspace $ \{\ langle V_ I \ rangle, 1 \ Leq I \ Leq r \} $ straight sum:

\ [V = \ langle v_1 \ rangle \ oplus \ langle V_2 \ rangle \ oplus \ cdots \ oplus \ langle v_r \ rangle. \]

This decomposition method is uniquely identified in the following sense: If

\ [V = \ langle W_1 \ rangle \ oplus \ langle W_2 \ rangle \ oplus \ cdots \ oplus \ langle v_s \ rangle. \]

Is another way of decomposing it into cyclic subspaces. The order of $ \ {W_1, \ ldots, and w_s \} $ is also the power of the prime polynomial $ \ {Q_1 (X) ^ {F_1}, \ ldots, q_s (x) ^ {f_s }\}$, then $ r = S $ and Two Sets

\ [\ {P_1 (x) ^ {E_1}, \ ldots, P_r (X) ^ {e_r }\}\ quad \ Text {And} \ quad \ {Q_1 (x) ^ {F_1}, \ ldots, q_s (X) ^ {f_s }\}\]

There is only one difference.

Proof: the first step to prove rational standard form theorem is to use quasi-prime decomposition, as long as the minimum polynomial $ M_a (x) = p (x) of $ A $) ^ A $ is the power of a prime polynomial.

We assert that the order of $ V \ In V $ is exactly $ p (x) ^ A $. Otherwise, the order of any $ V $ is a factor of $ p (x) ^ A $, but not equal to $ p (x) ^ A $, and $ p (x) ^ {A-1} $ factor, thus $ P (a) ^ {A-1} = 0 $, with the smallest polynomial of $ A $ is $ p (x) ^ A $ conflict.

Therefore, take $ v_1 $ and set the order of $ v_1 $ to $ p (x) ^ A $. Then $ v_1 $ generates a cyclic sub-space $ \ langle v_1 \ rangle $. If $ \ langle v_1 \ rangle \ ne v $, we assert that we can find a $ A-$ constant sub-space $ W $ to make

\ [V = \ langle v_1 \ rangle \ oplus W. \]

In this way, we can continue this operation on $ W $, and finally break $ V $ into straight sums of some cyclic subspaces.

Consider the non-degraded bilinear type of $ V \ times V ^ \ ast \ rightarrow F $.

\ [(V, f) \ rightarrow F (V), \ quad V \ In V, f \ In V ^ \ ast. \]

Set $ A ^ \ ast $ to the adjoint transformation of $ A $: $ (AV, f) = (V, A ^ \ ast f) $, for any polynomial $ g (x) \ In F [x] $ (G (a) V, f) = (V, G (A ^ \ AST) F) $. Therefore, $ A $ and $ A ^ \ ast $ have the same extremely small polynomials.

Proof of concept: we do not know whether there is a breakdown in $ V $. $ v = \ langle v_1 \ rangle \ oplus W $, however, in $ V ^ \ ast $, $ \ langle v_1 \ rangle ^ \ bot $ is a constant sub-space of $ A ^ \ Ast-$, in $ V ^ \ ast $, find $ \ langle v_1 \ rangle ^ \ bot $ the constant sub-space $ W_1 $ to make $ V ^ \ ast = \ langle v_1 \ rangle ^ \ bot \ oplus W_1 $, $ v = \ langle v_1 \ rangle \ oplus W_1 ^ \ bot $ and $ W_1 ^ \ bot $ are available in $ V $!

Set $ \ langle v_1 \ rangle ^ \ bot $ to $ \ langle v_1 \ rangle $, which is a zero sub in $ V ^ \ ast $, then $ \ langle v_1 \ rangle ^ \ bot $ is the constant sub-space of $ A ^ \ ast $, therefore, $ A ^ \ ast $ has an inductive transformation on $ V ^ \ AST/\ langle v_1 \ rangle ^ \ bot $ \ widetilde {A ^ \ ast} $. We already know the limitations of $ A $ on $ \ langle v_1 \ rangle $ and $ \ widetilde {A ^ \ ast} $ on $ V ^ \ AST/\ langle v_1 \ rangle ^ \ bot $ is a pair of adjoint transformations, therefore, the extremely small polynomial of $ \ widetilde {A ^ \ ast} $ on $ V ^ \ AST/\ langle v_1 \ rangle ^ \ bot $ is also $ p (x) ^ A $, resulting in $ [f] \ In V ^ \ AST/\ langle v_1 \ rangle ^ \ bot $. The order of $ [f] $ is $ p (x) ^ A $.

Make $ W_1 $ \ {F, a ^ \ ast F, \ ldots, (a ^ \ AST) ^ {D-1} f \}$ refers to the sub-space in $ V ^ \ ast $, where $ d = \ deg p (x) ^ A $. Obviously, $ W_1 $ is a $ A ^ \ Ast-$ constant sub-space with a dimension of $ d $ and $ W_1 \ cap \ langle v_1 \ rangle ^ \ bot = 0 $.

Now $ \ dim \ langle v_1 \ rangle ^ \ bot = \ dim V ^ \ Ast-d $

\ [V ^ \ ast = \ langle v_1 \ rangle ^ \ bot \ oplus W_1, \]

This is a $ A ^ \ Ast-$ of $ V ^ \ ast $, which returns to $ V $ and returns to the following:

\ [V = \ langle v_1 \ rangle \ oplus (W_1) ^ \ bot. \]

In this way, the desired $ A-$ constant sub-space is obtained directly and decomposed.

Non-degraded bilinear linear type