This work by Jephian Lin is licensed under a Creative Commons Attribution 4.0 International License.
Direct sum of orthogonal subspaces
This work by Jephian Lin is licensed under a Creative Commons Attribution 4.0 International License.
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from lingeo import random_int_list, random_good_matrix, kernel_matrix
Let $U$ and $V$ be two subspaces under the same inner product space.
We say $U$ and $V$ are orthogonal if $\inp{\bu}{\bv} = 0$ for any $\bu\in U$ and $\bv\in V$.
Similarly, we say a collection of subspaces $\{V_1,\ldots,V_k\}$ is orthogonal if they are pairwisely orthogonal.
If $\{V_1,\ldots,V_k\}$ is orthogonal and none of them are $\{\bzero\}$, then they are necssarily independent.
Therefore, we have the direct sum $V_1 \oplus \cdots \oplus V_k$.
Suppose $V = V_1 \oplus \cdots \oplus V_k$.
Then every vector $\bv\in V$ can be uniquely written as $\bv = \bv_1 + \cdots + \bv_k$ with $\bv_i\in V_i$ for each $i = 1,\ldots,k$.
Let $A$ be an $m\times n$ matrix.
We have seen several cases of mutually orthogonal subspaces related to $A$.
With the new terminology, we may safely say:
Suppose $V$ is a subspace in $\mathbb{R}^n$.
We also have $\mathbb{R}^n = V \oplus V^\perp$.
執行以下程式碼。
已知 $R$ 為 $A$ 的最簡階梯形式矩陣。
Run the code below. Let $R$ be the reduced echelon form of $A$.
### code
set_random_seed(0)
print_ans = False
m,n,r = 2,4,2
A = random_good_matrix(m,n,r)
R = A.rref()
H = kernel_matrix(R)
c = random_int_list(2, r=3)
b = c[0]*R[0] + c[1]*H.transpose()[0]
print("A =")
show(A)
print("R =")
show(R)
print("b = ", b)
if print_ans:
r = c[0]*R[0]
h = c[1]*H.transpose()[0]
print("r =", r)
print("h =", h)
print("|b|^2 =", b.norm()**2)
print("|r|^2 + |h|^2 = %s + %s = %s"%(r.norm()**2, h.norm()**2, r.norm()**2 + h.norm()**2))
將 $\bb$ 寫成 $\br + \bh$
其中 $\br\in\Row(A)$ 而 $\bh\in\ker(A)$。
Write $\bb$ as $\br + \bh$ such that $\br\in\Row(A)$ and $\bh\in\ker(A)$.
證驗 $\br$ 和 $\bh$ 垂直﹐
而且 $\|\bb\|^2 = \|\br\|^2 + \|\bh\|^2$。
Verify that $\br$ and $\bh$ are orthogonal and $\|\bb\|^2 = \|\br\|^2 + \|\bh\|^2$.
因為每一個 $\mathbb{R}^n$ 中的向量都可以分解成 $\br\in\Row(A)$ 和 $\bh\in\ker(A)$ 中的向量相加。
說明對任何 $m\times n$ 矩陣都有
Since any vector in $\mathbb{R}^n$ can be written as the sum of vectors $\br\in\Row(A)$ and $\bh\in\ker(A)$. Show that
$$ \{A\bx: \bx\in\mathbb{R}^n \} = \{ A\br : \br\in\Row(A)\}. $$for any $m\times n$ matrix $A$.
令 $S = \{V_1,\ldots,V_k\}$ 為一群子空間。
證明若 $S$ 是垂直的集合且 $S$ 不包含 $\{\bzero\}$﹐則 $S$ 線性獨立。
Let $S = \{V_1,\ldots,V_k\}$ be a family of subspaces. Show that if $S$ is orthogonal and none of its elements is $\{\bzero\}$, then $S$ is linearly independent.
若 $S = \{V_1, V_2, V_3\}$ 垂直。
令 $V = V_1 \oplus V_2 \oplus V_3$、
$P$ 為 $V$ 的投影矩陣、
$P_1,P_2,P_3$ 分別為 $V_1,V_2,V_3$ 的投影矩陣。
Suppose $S = \{V_1,\ldots,V_k\}$ is an orthogonal set of subspaces. Let $V = V_1 \oplus V_2 \oplus V_3$, $P$ the projection matrix onto $V$, and $P_1, P_2, P_3$ the projection matrices onto $V_1, V_2, V_3$, respectively.
若 $V = \mathbb{R}^n$ 為全空間。
說明 $I_n = P_1 + P_2 + P_3$。
Suppose $V = \mathbb{R}^n$ is the whole space. Show that $I_n = P_1 + P_2 + P_3$.
依照步驟證明以下敘述等價：
Use the given instructions to show that the following are equivalent:
證明若 $P$ 是一個投影矩陣﹐
則 $P$ 是 $\Col(P)$ 的投影矩陣。
因此如果 $\bu\in\Col(P)$ 則 $P\bu = \bu$、
如果 $\bu\in\ker(P\trans)$ 則 $P\bu = \bzero$、
同時每個向量都可以分成 $\bu = P\bu + (I - P)\bu$
使得 $P\bu\in\Col(P)$ 且 $(I - P)\bu\in\ker(P\trans)$。
藉由這些性質說明如果條件一成立則條件二成立。
If $P$ is a projection matrix, then $P$ is the projection matrix onto $\Col(P)$. Therefore, $P\bu = \bu$ if $\bu\in\Col(P)$ and $P\bu = \bzero$ if $\bu\in\ker(P\trans)$. Also, every vector can be written as $\bu = P\bu + (I - P)\bu$ such that $P\bu\in\Col(P)$ and $(I - P)\bu\in\ker(P\trans)$.
Justify these statements and use them to show that Condition 1 implies Condition 2.
若 $P$ 對稱且 $P = P^2$。
說明 $\mathbb{R}^n = \Col(P) \oplus \ker(P)$ 且
如果 $\bu\in\Col(P)$ 則 $P\bu = \bu$、
如果 $\bu\in\ker(P)$ 則 $P\bu = \bzero$。
藉由這些性質說明如果條件二成立則條件一成立。
Suppose $P$ is a symmetric matrix and $P = P^2$. Then $\mathbb{R}^n = \Col(P) \oplus \ker(P)$. Also, $P\bu = \bu$ if $\bu\in\Col(P)$ and $P\bu = \bzero$ if $\bu\in\ker(P)$.
Justify these statements and use them to show that Condition 2 implies Condition 1.
證明若 $V = V_1 \oplus \cdots \oplus V_k$﹐
則每一個向量 $\bv\in V$ 都可以被寫成 $\bv = \bv_1 + \cdots + \bv_k$﹐
使得對每一個 $i = 1,\ldots,k$ 都有 $\bv_i\in V_i$﹐
而且這種寫法唯一。
Show that if $V = V_1 \oplus \cdots \oplus V_k$, then every vector $\bv\in V$ can be written as $\bv = \bv_1 + \cdots + \bv_k$ such that $\bv_i\in V_i$ for each $i = 1, \ldots, k$. Moreover, such a representation is unique.
利用垂直空間分解母空間的現象在其它向量空間也很常見。
It is also common to see other vector spaces being decomposed into orthogonal subspaces.
令 $U = \mathcal{M}_{n\times n}$ 為一向量空間，搭配內積 $\inp{A}{B} = \tr(B\trans A)$。
令 $V$ 為 $U$ 中所有對稱矩陣的集合、
令 $W$ 為 $U$ 中所有反對稱矩陣的集合。
說明 $\{V, W\}$ 垂直且 $U = V \oplus W$。
Let $U = \mathcal{M}_{n\times n}$ be the vector space equipped with the inner product $\inp{A}{B} = \tr(B\trans A)$. Let $V$ be the subspace of $U$ consisting of all symmetric matrices. Let $W$ be the subspace of $U$ consisting of all skew-symmetric matrices. Show that $\{V, W\}$ is orthogonal and $U = V \oplus W$.
令 $U = \mathcal{P}_{d}$ 為一向量空間，搭配內積
$$ \inp{a_0 + a_1x + \cdots + a_dx^d}{b_0 + b_1x + \cdots + b_dx^d} = a_0b_0 + a_1b_1 + \cdots + a_db_d. $$令 $V$ 為 $U$ 中所有偶函數的集合、
令 $W$ 為 $U$ 中所有奇函數的集合。
說明 $V$ 和 $W$ 垂直且 $U = V \oplus W$。
Let $U = \mathcal{P}_{d}$ be the vector space equipped with the inner product
$$ \inp{a_0 + a_1x + \cdots + a_dx^d}{b_0 + b_1x + \cdots + b_dx^d} = a_0b_0 + a_1b_1 + \cdots + a_db_d. $$Let $V$ be the subspace of $U$ consisting of all even functions. Let $W$ be the subspace of $U$ consisting of all odd functions. Show that $\{V, W\}$ is orthogonal and $U = V \oplus W$.