## 数学代写|线性代数代写linear algebra代考|MATHS1011

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## 数学代写|线性代数代写linear algebra代考|The Contrast Between Nul A and Col A

It is natural to wonder how the null space and column space of a matrix are related. In fact, the two spaces are quite dissimilar, as Examples $5-7$ will show. Nevertheless, a surprising connection between the null space and column space will emerge in Section 4.6, after more theory is available.
EXAMPLE 5 Let
$$A=\left[\begin{array}{rrrr} 2 & 4 & -2 & 1 \ -2 & -5 & 7 & 3 \ 3 & 7 & -8 & 6 \end{array}\right]$$
a. If the column space of $A$ is a subspace of $\mathbb{R}^k$, what is $k$ ?
b. If the null space of $A$ is a subspace of $\mathrm{R}^k$, what is $k$ ?
SOLUTION
a. The columns of $A$ each have three entries, so $\operatorname{Col} A$ is a subspace of $\mathbb{R}^k$, where $k=3$.
b. A vector $\mathbf{x}$ such that $A \mathbf{x}$ is defined must have four entries, so $\mathrm{Nul} A$ is a subspace of $\mathbb{R}^k$, where $k=4$.

When a matrix is not square, as in Example 5, the vectors in $\mathrm{Nul} A$ and $\operatorname{Col} A$ live in entirely different “universes.” For example, no linear combination of vectors in $\mathbb{R}^3$ can produce a vector in $\mathbb{R}^4$. When $A$ is square, $\operatorname{Nul} A$ and $\operatorname{Col} A$ do have the zero vector in common, and in special cases it is possible that some nonzero vectors belong to both $\operatorname{Nul} A$ and $\operatorname{Col} A$.

## 数学代写|线性代数代写linear algebra代考|LINEARLY INDEPENDENT SETS; BASES

In this section we identify and study the subsets that span a vector space $V$ or a subspace $H$ as “efficiently” as possible. The key idea is that of linear independence, defined as in $\mathbb{R}^n$.

An indexed set of vectors $\left{\mathbf{v}_1, \ldots, \mathbf{v}_p\right}$ in $V$ is said to be linearly independent if the vector equation
$$c_1 \mathbf{v}_1+c_2 \mathbf{v}_2+\cdots+c_p \mathbf{v}_p=\mathbf{0}$$
has only the trivial solution, $c_1=0, \ldots, c_p=0^1$
The set $\left{\mathbf{v}_1, \ldots, \mathbf{v}_p\right}$ is said to be linearly dependent if $(1)$ has a nontrivial solution, that is, if there are some weights, $c_1, \ldots, c_p$, not all zero, such that (1) holds. In such a case, (1) is called a linear dependence relation among $\mathbf{v}_1, \ldots, \mathbf{v}_p$.

Just as in $\mathbb{R}^n$, a set containing a single vector $\mathbf{v}$ is linearly independent if and only if $\mathbf{v} \neq \mathbf{0}$. Also, a set of two vectors is linearly dependent if and only if one of the vectors is a multiple of the other. And any set containing the zero vector is linearly dependent. The following theorem has the same proof as Theorem 7 in Section 1.7.
An indexed set $\left{\mathbf{v}1, \ldots, \mathbf{v}_p\right}$ of two or more vectors, with $\mathbf{v}_1 \neq \mathbf{0}$, is linearly dependent if and only if some $\mathbf{v}_j$ (with $j>1$ ) is a linear combination of the preceding vectors, $\mathbf{v}_1, \ldots, \mathbf{v}{j-1}$.
The main difference between linear dependence in $\mathbb{R}^n$ and in a general vector space is that when the vectors are not $n$-tuples, the homogeneous equation (1) usually cannot be written as a system of $n$ linear equations. That is, the vectors cannot be made into the columns of a matrix $A$ in order to study the equation $A \mathbf{x}=\mathbf{0}$. We must rely instead on the definition of linear dependence and on Theorem 4 .

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|The Contrast Between Nul A and Col A

$$A=\left[\begin{array}{llllllllllll} 2 & 4 & -2 & 1 & -2 & -5 & 7 & 3 & 3 & 7 & -8 & 6 \end{array}\right]$$

b. 如果零空间 $A$ 是一个子空间 $\mathrm{R}^k$ ，什么是 $k$ ?

b. 向量 $\mathbf{x}$ 文样 $A \mathbf{x}$ 被定义必须有四个条目，所以Nul $A$ 是 一个子空间 $\mathbb{R}^k$ ， 在哪里 $k=4$.

$\mathrm{Nul} A$ 和Col $A$ 确实有共同的零向量，在特殊情况下，

## 数学代写|线性代数代写linear algebra代考|LINEARLY INDEPENDENT SETS; BASES

Ueft{\mathbf{v}_1, Vdots, Imathbf{v}_plright $}$ 在 $V$ 据说是线生无关的，如果向量方程
$$c_1 \mathbf{v}_1+c_2 \mathbf{v}_2+\cdots+c_p \mathbf{v}_p=\mathbf{0}$$

$\mathbf{v}_1, \ldots, \mathbf{v} j-1$

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## MATLAB代写

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