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

5.9.1 Consider the function $F: \mathbb{R}^2 \longrightarrow \mathbb{R}^2$ defined by: $F(x, y)=(x+2 k y, x-y)$. Determine the values of $k$ for which $F$ is linear.
5.9.2 Given the function $F: \mathbb{R}_1[x] \longrightarrow \mathbb{R}_2[x]$ defined by: $F(a x+b)=(a-b) x^2+$ $k b^2 x+2 a$. Determine the values of $k$ for which $F$ is linear.

5.9.3 Given linear transformations $F: \mathbb{R}^3 \rightarrow \mathbb{R}^2$ and $G: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ defined by: $F(x, y, z)=(x-y, 2 x+y+z)$ and $G(x, y)=(3 y,-x, 4 x+2 y)$, determine if possible, $F \circ G$ and $G \circ F$.
5.9.4 Given the linear transformations $F: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ and $G: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined by: $F\left(\mathbf{e}_1\right)=-\mathbf{e}_1-\mathbf{e}_2, F\left(\mathbf{e}_2\right)=\mathbf{e}_1+\mathbf{e}_2, G\left(\mathbf{e}_1\right)=2, G\left(\mathbf{e}_2\right)=-1$; determine if possible $F \circ G$ and $G \circ F$.
5.9.5 Consider the linear transformation $F: \mathbb{R}^2 \longrightarrow \mathbb{R}^2$ defined by $F\left(\mathbf{e}_1\right)=3 \mathbf{e}_1-3 \mathbf{e}_2$, $F\left(\mathbf{e}_2\right)=2 \mathbf{e}_1-2 \mathbf{e}_2$. Compute a basis of the kernel and a basis for the image of $F$.
5.9.6 Establish which of the following linear transformations are isomorphisms:
i) $F: \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ defined by $F(x, y, z)=(x+2 z, y+z, z)$;
ii) $F: \mathbb{R}^3 \rightarrow \mathbb{R}^2$ defined by $F(x, y, z)=(2 x-z, x-y+z)$;
iii) $F: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ defined by $F\left(\mathbf{e}_1\right)=2 \mathbf{e}_1+\mathbf{e}_2, F\left(\mathbf{e}_2\right)=3 \mathbf{e}_1-\mathbf{e}_3, F\left(\mathbf{e}_3\right)=\mathbf{e}_1-\mathbf{e}_2-\mathbf{e}_3$.
5.9.7 Find a basis for the kernel and one for the image of each of the following linear transformations. Establish if they are injective, surjective and/or bijective.
i) $F: \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ defined by $F(x, y, z)=(x-z, x+2 y-z, x-4 y-z)$.
ii) $F: \mathbb{R}^2 \longrightarrow \mathbb{R}^3$ defined by $F\left(\mathbf{e}_1\right)=\mathbf{e}_1+\mathbf{e}_2-\mathbf{e}_3, F\left(\mathbf{e}_2\right)=2 \mathbf{e}_1-2 \mathbf{e}_2-\mathbf{e}_3$.
iii) $F: \mathbb{R}^4 \longrightarrow \mathbb{R}^2$ definer hy $F(x, y, z, t)=(2 x-t, 3 y-x+2 z-t)$

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

In Chapter 5, we defined the row rank of a matrix (see Definition 5.7.3). We now want to deepen the study of this notion and have a clearer view of the link between matrices, linear systems and transformations.

Given a matrix $A \in \mathrm{M}{m, n}(\mathbb{R})$, we can read its rows as vectors of $\mathbb{R}^n$ and its columns as vectors of $\mathbb{R}^m$. It is therefore natural to introduce the following definition. Definition 6.2.1 We call column rank of a matrix $A \in \mathrm{M}{m, n}(\mathbb{R})$, the maximum number of linearly independent columns of $A$, i.e. the dimension of the subspace of $\mathbb{R}^m$ generated by the columns of $A$.

The following observation is already known and yet, given the its importance in the context that we are studying, we want to rexamine it.

Observation 6.2.2 If we write $A$ as the matrix associated with the linear transformation $L_A: \mathbb{R}^n \longrightarrow \mathbb{R}^m$ with respect to the canonical bases, then the column rank of $A$ is the dimension of the image of $L_A$. Indeed, the image is generated by the columns of the matrix $A$.

Although in general the row vectors and column vectors of a matrix $A \in \mathrm{M}_{m, n}(\mathbb{R})$ are elements of different vector spaces, the row and column rank of $A$ always coincide. This number is simply called $\operatorname{rank}$ of $A$, denoted by $\operatorname{rk}(A)$.

Proposition 6.2.3 If $A \in \mathrm{M}_{m, n}(\mathbb{R})$, then the row rank of $A$ is equal to the column rank of $A$.

# 线性代数代考

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

5.9.1 考虑函数 $F: \mathbb{R}^2 \longrightarrow \mathbb{R}^2$ 被定义为: $F(x, y)=(x+2 k y, x-y)$. 确定的值 $k$ 为此 $F$ 是线性的。
5.9.2 给定函数 $F: \mathbb{R}_1[x] \longrightarrow \mathbb{R}_2[x]$ 被定义为: $F(a x+b)=(a-b) x^2+k b^2 x+2 a$. 确定的值 $k$ 为此 $F$ 是线性的。
$5.9 .3$ 给定的线性变换 $F: \mathbb{R}^3 \rightarrow \mathbb{R}^2$ 和 $G: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ 被定义为: $F(x, y, z)=(x-y, 2 x+y+z)$ 和 $G(x, y)=(3 y,-x, 4 x+2 y)$ ，确定是否可能, $F \circ G$ 和 $G \circ F$.
$5.9 .4$ 给定线性变换 $F: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ 和 $G: \mathbb{R}^2 \rightarrow \mathbb{R}$ 被定义为:
$F\left(\mathbf{e}_1\right)=-\mathbf{e}_1-\mathbf{e}_2, F\left(\mathbf{e}_2\right)=\mathbf{e}_1+\mathbf{e}_2, G\left(\mathbf{e}_1\right)=2, G\left(\mathbf{e}_2\right)=-1$; 确定是否可能 $F \circ G$ 和 $G \circ F$.
$5.9 .5$ 考虑线性变换 $F: \mathbb{R}^2 \longrightarrow \mathbb{R}^2$ 被定义为 $F\left(\mathbf{e}_1\right)=3 \mathbf{e}_1-3 \mathbf{e}_2, F\left(\mathbf{e}_2\right)=2 \mathbf{e}_1-2 \mathbf{e}_2$. 计算核的基和图 像的基 $F$.
5.9.6 确定下列哪些线性变换是同构的:
i) $F: \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ 被定义为 $F(x, y, z)=(x+2 z, y+z, z)$;
ii) $F: \mathbb{R}^3 \rightarrow \mathbb{R}^2$ 被定义为 $F(x, y, z)=(2 x-z, x-y+z)$;
iii) $F: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ 被定义为 $F\left(\mathbf{e}_1\right)=2 \mathbf{e}_1+\mathbf{e}_2, F\left(\mathbf{e}_2\right)=3 \mathbf{e}_1-\mathbf{e}_3, F\left(\mathbf{e}_3\right)=\mathbf{e}_1-\mathbf{e}_2-\mathbf{e}_3$.
$5.9 .7$ 为以下每个线性变换的核和图像找到一个基。确定它们是单射的、满射的和/或双射的。

ii) $F: \mathbb{R}^2 \longrightarrow \mathbb{R}^3$ 被定义为 $F\left(\mathbf{e}_1\right)=\mathbf{e}_1+\mathbf{e}_2-\mathbf{e}_3, F\left(\mathbf{e}_2\right)=2 \mathbf{e}_1-2 \mathbf{e}_2-\mathbf{e}_3$.
iii) $F: \mathbb{R}^4 \longrightarrow \mathbb{R}^2$ 他定义 $F(x, y, z, t)=(2 x-t, 3 y-x+2 z-t)$

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

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