# 数学代写|线性代数代写linear algebra代考|Row and column vectors

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## 数学代写|线性代数代写linear algebra代考|Row and column vectors

What are the ‘row vectors of a matrix’?
The row vectors of a matrix are the entries in the rows of a given matrix. For example, the row vectors of $\mathbf{A}=\left(\begin{array}{lll}1 & 2 & 3 \ 4 & 5 & 6\end{array}\right)$ are $\left(\begin{array}{l}1 \ 2 \ 3\end{array}\right)$ and $\left(\begin{array}{l}4 \ 5 \ 6\end{array}\right)$ because the first row of matrix $\mathbf{A}$ has the entries 1,2 and 3 and the second row has entries 4,5 and 6.
What are the column vectors of $\mathrm{A}=\left(\begin{array}{lll}1 & 2 & 3 \ 4 & 5 & 6\end{array}\right)$ ?
The matrix $\mathbf{A}$ has three columns vectors:
$$\left(\begin{array}{l} 1 \ 4 \end{array}\right),\left(\begin{array}{l} 2 \ 5 \end{array}\right) \text { and }\left(\begin{array}{l} 3 \ 6 \end{array}\right)$$
We generalize this by considering the $m$ by $n$ matrix:
What are the row vectors of this matrix $\mathbf{A}$ ?
The row vectors of A denoted $\mathbf{r}1, \mathbf{r}_2, \ldots$ and $\mathbf{r}_m$ are given by: $$\mathbf{r}_1=\left(\begin{array}{c} a{11} \ a_{12} \ \vdots \ a_{1 n} \end{array}\right), \quad \mathbf{r}2=\left(\begin{array}{c} a{21} \ a_{22} \ \vdots \ a_{2 n} \end{array}\right), \cdots \text { and } \mathbf{r}m=\left(\begin{array}{c} a{m 1} \ a_{m 2} \ \vdots \ a_{m n} \end{array}\right)$$

## 数学代写|线性代数代写linear algebra代考|Row and column space

The row space of a matrix $\mathbf{A}$ is the space spanned by the row vectors of $\mathbf{A}$. Remember, the space spanned by vectors means the space containing all the linear combinations of these vectors.
What is the row space of the above matrix $\mathbf{B}$ ?
It is the space, $S$, spanned by the vectors $\mathbf{r}_1=\left(\begin{array}{r}-3 \ 6\end{array}\right), \mathbf{r}_2=\left(\begin{array}{r}-5 \ 2\end{array}\right)$ and $\mathbf{r}_3=\left(\begin{array}{r}-2 \ 7\end{array}\right)$.
Any linear combination of these vectors, $\mathbf{r}_1, \mathbf{r}_2$ and $\mathbf{r}_3$, is in the row space $S$. Hence
$$\text { Row Space } S=\operatorname{span}\left{\mathbf{r}_1, \mathbf{r}_2, \mathbf{r}_3\right}=k_1\left(\begin{array}{r} -3 \ 6 \end{array}\right)+k_2\left(\begin{array}{r} -5 \ 2 \end{array}\right)+k_3\left(\begin{array}{r} -2 \ 7 \end{array}\right)$$
where $k_1, k_2$ and $k_3$ are scalars. Each row vector has two entries of real numbers and we can show that $S$ is a subspace of $\mathbb{R}^2$.

The row space $S$ is the set of vectors $\mathbf{u}$ such that $\mathbf{u}=k_1 \mathbf{r}_1+k_2 \mathbf{r}_2+k_3 \mathbf{r}_3$. This row space $S$, spanned by $\mathbf{r}_1, \mathbf{r}_2$ and $\mathbf{r}_3$, is the vector space given by
$$S=\left{\mathbf{u} \mid \mathbf{u}=k_1 \mathbf{r}_1+k_2 \mathbf{r}_2+k_3 \mathbf{r}_3\right}$$
In this case, these vectors span the whole of $\mathbb{R}^2$ because no two vectors in $S$ are multiplies of each other (they are linearly independent). The row space of the above matrix $\mathbf{B}$ occupies $\mathbb{R}^2$. Similarly the column space of a general matrix $\mathbf{A}$ is the space spanned by the column vectors of $\mathbf{A}$. We formally define the row and column space as follows:
Definition (3.23). Let $\mathbf{A}$ be any matrix. Then
(a) The row space of the matrix $\mathbf{A}$ is the space spanned by the row vectors of matrix $\mathbf{A}$.
(b) The column space of the matrix $\mathbf{A}$ is the space spanned by the column vectors of matrix $\mathbf{A}$.

# 线性代数代考

## 数学代写|线性代数代写linear algebra代考|Row and column vectors

$\mathrm{A}=\left(\begin{array}{lll}1 & 2 & 3 \ 4 & 5 & 6\end{array}\right)$的列向量是什么?

$$\left(\begin{array}{l} 1 \ 4 \end{array}\right),\left(\begin{array}{l} 2 \ 5 \end{array}\right) \text { and }\left(\begin{array}{l} 3 \ 6 \end{array}\right)$$

A的行向量表示$\mathbf{r}1, \mathbf{r}2, \ldots$和$\mathbf{r}_m$为: $$\mathbf{r}_1=\left(\begin{array}{c} a{11} \ a{12} \ \vdots \ a_{1 n} \end{array}\right), \quad \mathbf{r}2=\left(\begin{array}{c} a{21} \ a_{22} \ \vdots \ a_{2 n} \end{array}\right), \cdots \text { and } \mathbf{r}m=\left(\begin{array}{c} a{m 1} \ a_{m 2} \ \vdots \ a_{m n} \end{array}\right)$$

## 数学代写|线性代数代写linear algebra代考|Row and column space

$$\text { Row Space } S=\operatorname{span}\left{\mathbf{r}_1, \mathbf{r}_2, \mathbf{r}_3\right}=k_1\left(\begin{array}{r} -3 \ 6 \end{array}\right)+k_2\left(\begin{array}{r} -5 \ 2 \end{array}\right)+k_3\left(\begin{array}{r} -2 \ 7 \end{array}\right)$$

$$S=\left{\mathbf{u} \mid \mathbf{u}=k_1 \mathbf{r}_1+k_2 \mathbf{r}_2+k_3 \mathbf{r}_3\right}$$

(a)矩阵$\mathbf{A}$的行空间是矩阵$\mathbf{A}$的行向量张成的空间。
(b)矩阵$\mathbf{A}$的列空间是矩阵$\mathbf{A}$的列向量张成的空间。

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