Subspaces in a vector space
This work by Jephian Lin is licensed under a Creative Commons Attribution 4.0 International License.
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Let $(U, +, \cdot)$ be a vector space.
If $V\subseteq U$ along with the same $+$ and $\cdot$ is also a vector space, then naturally $V$ is a substructure under $U$.
Since most of the good properties are inherited from $U$, we only need to check the closedness and the existence of the additive identity.
(Notice that $V$ has to use the same operations.)
Let $U$ be a vector space and $V\subseteq U$.
Then $V$ is a subspace of $U$ under the same vector addition and scalar multiplication if:
A set $V$ is a subspace of the vector space $U$ if and only if $V = \vspan(S)$ for some $S\subseteq U$.
若 $U$ 是一個向量空間
而 $V$ 是一個 $U$ 中的子空間。
證明 $V$ 也是一個向量空間。
Let $U$ be a vector space and $V$ a subspace of $U$. Show that $V$ is also a vector space.
令 $U$ 為一個向量空間且 $S\subseteq U$。
回顧 $\mathbb{R}^n$ 中的各種性質。
試著寫出下列概念的定義。
Let $U$ be a vector space and $S\subseteq U$. Based on the properties in $\mathbb{R}^n$, guess the definition of the following concepts.
寫出 $S$ 的線性組合以及 $\vspan(S)$ 的定義。
What is a linear combination of $S$? What is $\vspan(S)$?
寫出 $S$ 是一個子空間 $V$ 的基底的定義、
並定義 $V$ 的維度。
What is the meaning of "$S$ is a basis of $V$"? Define the dimension of $V$.
令 $U$ 為所有三次以下的多項式。
考慮 $S$ 為所有 $U$ 中可以寫成 $(x-1)p(x)$ 的多項式的集合。
Let $U$ be the set of all polynomials of degree at most $3$. Let $S$ be the subset of $U$ consisting of all polynomials of the form $(x-1)p(x)$ for some polynomial $p(x)$.
找出 $S$ 的一組基底﹐並算出 $S$ 的維度。
(可以先找出一組生成集﹐再判斷其是否獨立。)
Find a basis of $S$ and determine the dimension of $S$. (Try to find a spanning set first and then show it is linearly independent.)
令 $U$ 為所有 $3\times 3$ 的矩陣。
考慮 $S$ 為所有 $U$ 中的上三角矩陣
(所有非零項都在對角線及其上方的部位內)。
Let $U$ be the set of all $3\times 3$ matrices. Let $S$ be the subset of $U$ consisting of all upper triangular matrices. (That is, an upper triangular matrix is a matrix whose nonzero entries only occurs on the diagonal or above.)
找出 $S$ 的一組基底﹐並算出 $S$ 的維度。
(可以先找出一組生成集﹐再判斷其是否獨立。)
Find a basis of $S$ and determine the dimension of $S$. (Try to find a spanning set first and then show it is linearly independent.)
令 $\mathcal{P}_3$ 為所有三次以下多項式所形成的集合。
從中取出兩個多項式 $x^3$ 和 $x^3 + 1$﹐
由於 $x^3\cdot(x^3 + 1)$ 是六次式﹐
$\mathcal{P}_3$ 對乘法不封閉﹐
因此不是一個子空間。
Let $\mathcal{P}_3$ be the set of all polynomials of degree at most $3$. We may choose two polynomials, e.g., $x^3$ and $x^3 + 1$ such that their product $x^3\cdot(x^3 + 1)$ has degree $6$. Therefore, $\mathcal{P}_3$ is not closed under multiplication and it is not a vector space.
令 $\operatorname{Sym}_2(\mathbb{R})$ 為所有 $2\times 2$ 的實對稱矩陣所形成的集合。
從中取出兩個矩陣
$A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$ 和
$B = \begin{bmatrix} 0 & 1 \\ 1 & 2 \end{bmatrix}$﹐
由於 $AB$ 不對稱﹐
$\operatorname{Sym}_2(\mathbb{R})$ 對乘法不封閉﹐
因此不是一個子空間。
Let $\operatorname{Sym}_2(\mathbb{R})$ be the set of all $2\times 2$ real symmetric matrices. We may choose two matrices, e.g., $A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 1 \\ 1 & 2 \end{bmatrix}$ such that their product $AB$ is not symmetric. Therefore, $\mathcal{P}_3$ is not closed under multiplication and it is not a vector space.
令 $U$ 為所有 $n\times n$ 的實數矩陣所形成的集合。
令 $V$ 為 $U$ 中的對稱矩陣的集合。
令 $W$ 為 $U$ 中的反對稱矩陣的集合。
(一個矩陣 $A$ 如果有 $A\trans = -A$﹐則被稱為反對稱。)
證明每一個 $U$ 中的元素 $\bb$
都可以寫成 $\bb = \bv + \bw$
使得 $\bv\in V$ 且 $\bw\in W$。
Let $U$ be the set of all $n\times n$ real matrices. Let $V$ be the subset of $U$ consisting of all symmetric matrices. Let $W$ be the subset of $U$ consisting of all skew-symmetric matrices. (A matrix $A$ is skew-symmetric if $A\trans = -A$.) Show that every matrix $\bb\in U$ can be written as $\bb = \bv + \bw$ such that $\bv\in V$ and $\bw\in W$.
令 $U$ 為所有 $10$ 次以下的多項式所形成的集合。
令 $V$ 為 $U$ 中的偶函數的集合。
令 $W$ 為 $U$ 中的奇函數的集合。
(一個多項式如果只有在偶數次方有非零係數﹐則被稱為偶函數。
一個多項式如果只有在奇數次方有非零係數﹐則被稱為奇函數。)
證明每一個 $U$ 中的元素 $\bb$
都可以寫成 $\bb = \bv + \bw$
使得 $\bv\in V$ 且 $\bw\in W$。
Let $U$ be the set of all polynomials of degree at most $10$. Let $V$ be the subset of $U$ consisting of all even functions. Let $W$ be the subset of $U$ consisting of all odd functions. (A polynomial is an even function if its nonzero coefficients only occurs at the positions of even degrees. A polynomial is an odd function if its nonzero coefficients only occurs at the positions of odd degrees.) Show that every polynomial $\bb\in U$ can be written as $\bb = \bv + \bw$ such that $\bv\in V$ and $\bw\in W$.
令 $U$ 為所有 $\mathbb{R}\rightarrow\mathbb{R}$ 的函數所形成的集合。
令 $V$ 為 $U$ 中的偶函數的集合。
令 $W$ 為 $U$ 中的奇函數的集合。
(一個函數 $f$ 如果對所有 $x$ 都有 $f(-x) = f(x)$﹐則被稱為偶函數。
一個函數 $f$ 如果對所有 $x$ 都有 $f(-x) = -f(x)$﹐則被稱為奇函數。)
證明每一個 $U$ 中的元素 $\bb$
都可以寫成 $\bb = \bv + \bw$
使得 $\bv\in V$ 且 $\bw\in W$。
Let $U$ be the set of all functions from $\mathbb{R}$ to $\mathbb{R}$. Let $V$ be the subset of $U$ consisting of all even functions. Let $W$ be the subset of $U$ consisting of all odd functions. (A function $f$ is an even function if $f(-x) = f(x)$ for all $x$. A function $f$ is an odd function if $f(-x) = -f(x)$ for all $x$.) Show that every function $\bb\in U$ can be written as $\bb = \bv + \bw$ such that $\bv\in V$ and $\bw\in W$.