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Exercise 1: Typical Questions about Linear Maps

A linear transformation $f:\reel^4 \rightarrow\reel^3$ has the following mapping matrix with respect to the standard bases in $\reel^4$ and $\reel^3\,:$

$$ \mF =\matind eFe =\begin{matr}{cccc} 1 & 1 & 2 & 1 \\\\ 3 & 0 & 3 & 3 \\\\ -1 & 2 & 1 & -1 \end{matr}. $$
A

Decide by direct matrix-vector product which of the vectors

$$\,\mathbf{u}_1=(1,-1,0,2)\,,\,\mathbf{u}_2=(-1,0,0,1)\,\,\,\,\mathrm{and} \,\,\,\,\mathbf{u}_3=(-1,-2,2,-1)\,$$

that belong to the kernel for $f$.

B

Decide without computations whether the vector $\,\mb=(2,9,-5)\,$ belongs to the range $f(\reel^4)$.

C

Find the dimension of the image space $f(\reel^4)$.

D

Find without further computations the dimension of ker$(f)\,.$

E

State without further computations a basis for $\ker(f)$.

F

State without further computations a basis for $\,f(\reel^4)\,.$

G

State without further computations the solution to the vector equation

$$\,f(\mx)=\mb=(2,9,-5)\,.$$


Exercise 2: Conclusions about the Reduced Mapping Matrix

About the mapping matrix $\,\mF =\matind eFe\,$ for a linear transformation $f:\reel^3\rightarrow\reel^3$ it is given:

$$\mathrm{trap}(\mF)=\begin{matr}{rrr} 1 & 0 & 3 \\\\ 0 & 1 & 1 \\\\ 0 & 0 & 0 \end{matr}\,.$$
H

Read from this a basis for ker$(f)\,$ and state the dimension of the range $f(\reel^3)$.

I

Is it also possible to determine a basis for the range?


Exercise 3: New Mapping Matrix from Change of Base

If you change the basis, often you can find a mapping matrix that is more simple and therefore easier to work with. We shall see an example of this here where we first start by some exercises in change of base.

In the vector space $\,\reel^2\,$ we consider the standard base $\,e=(\,(1,0),(0,1)\,)\,.$ A new basis $\,a=(\ma_1,\ma_2)\,$ for $\,\reel ^2\,$ are determined by

$$\ma_1 = (1,2)\,\,\,\,\mathrm{and}\,\,\,\,\ma_2 = (3,7)\,.$$
J

State the basis shift matrix $\,\matind eMa\,$ that shifts from $a$-coordinates to $e$-coordinates. A vector $\,\mv\,$ has with respect to the basis $a$ the coordinate matrix $\,\vekind av= \begin{matr}{r} -1 \\ 1 \end{matr}\,.$ Determine the coordinate matrix for $\,\mv\,$ with respect to the basis $e\,.$

K

State the basis shift matrix $\,\matind aMe\,$ that shifts from $e$-coordinates to $a$-coordinates. A vector $\,\mv\,$ has with respect to basis $e$ the coordinate matrix $\,\vekind ev= \begin{matr}{r} 2 \\ 3 \end{matr}\,.$ Determine the coordinate matrix for $\,\mv\,$ with respect to the basis $a\,.$

Let $f:\reel ^2\rightarrow\reel ^2$ be a linear transformation that with respect to the standard $e$-basis in $\reel ^2$ has the mapping matrix

$$ \matind eFe = \begin{matr}{rr} -1 & 1 \\\\ -4 & 3 \end{matr}. $$
L

Determine the mapping matrix for $f$ with respect to the basis $a\,.$

M

A vector $\,\mv\,$ has with respect to the basis $a$ the coordinate vector $\,\vekind av= \begin{matr}{r} m \\ n \end{matr}\,.$ Determine the coordinate vector for $\,f(\mv)\,$ with respect to the basis $a\,.$


Exercise 4: Linear Transformation in Abstract Vector Spaces

In this Exercise we work with abstract vector spaces. Thus, we do not know whether it concerns number spaces, matrix spaces polynomial spaces or you name it. This does not prevent us from investigating a linear transfomation that maps vectors in the one vector space onto vectors in the other vector space.

A 2-dimensional vector space $V$ has a basis $a=(\ma_1,\ma_2)\,$, and a 3-dimensional vector space $W$ has a basis $c=(\mc_1,\mc_2,\mc_3)\,$. A linear transformation $\,f:V\rightarrow W\,$ is given by

$$\,f(\ma_1)=\mc_1-2\mc_2+\mc_3\,\,\,\,\mathrm{and}\,\,\,\,f(\ma_2)=-2\mc_1+4\mc_2-2\mc_3\,.$$
N

State the mapping matrix $\,\matind cFa\,$, and find the image $\,\mathbf y\,$ of the vector $\,\mathbf x=3\ma_1-\ma_2\,$ using the mapping matrix.

O

Which of the vectors $\,\mathbf \ma_1+2\ma_2\,$ and $\,\mathbf 2\ma_1+\ma_2\,$ belong to the kernel for $f\,$? Solve the exercise without determining all of the kernel.

P

Determine (readily without making new computations) a basis for the kernel for $f\,$.

Q

Which of the vectors $\,\mc_1-2\mc_2+\mc_3\,$ and $\,2\mc_1-\mc_2+2\mc_3\,$ belong to $f(V)\,$?

R

State a basis for the range for $f\,$.


Exercise 5: Linear Mapping and Change of Base. Maple

In $\,\reel^3\,$ we are given the vectors

$$\mv_1=(1,2,0), \mv_2=(0,1,4)\,\,\,\,\mathrm{and}\,\,\,\,\mv_3=(0,0,1)$$

and in $\,\reel^4\,$ we are given the vectors

$$\mw_1=(1,0,0,0), \mw_2=(1,1,0,0), \mw_3=(1,1,1,0)\,\,\,\,\mathrm{and}\,\,\,\,\mw_4=(1,1,1,1)\,.$$
S

Show that the set $\,v=(\mv_1,\mv_2,\mv_3)\,$ constitutes a basis for $\,\reel^3\,$ and that the set $\,w=(\mw_1,\mw_2,\mw_3,\mw_4)\,$ constitutes a basis for $\,\reel^4\,.$

Now let $f:\reel^3\rightarrow\reel^4$ be the linear transformation, determined by

$$ \begin{aligned} f(\mv_1)=\mw_1+\mw_2,\\\\ f(\mv_2)=\mw_2+\mw_3,\\\\ f(\mv_3)=\mw_3+\mw_4. \end{aligned}$$
T

State the mapping matrix for $f$ with respect to the basis $v$ in $\reel^3$ and the basis $w$ in $\reel^4$.

U

Determine the mapping matrix $f$ with respect to the ordinary bases in $\reel^3$ and $\reel^4$, respectively.


Exercise 6: Extra Training Exercise in Linear Mappings

Let $(\mathbf{e}_1,\mathbf{e}_2)$ denote the standard basis for $\reel ^2$ and let $c=(\mathbf{c}_1,\mathbf{c}_2,\mathbf{c}_3,\mathbf{c}_4)$ denote some given basis for $\reel ^4$. Now let $f:\reel ^2\rightarrow\reel ^4$ be a linear transformation, where

$$ f(\mathbf{e}_1)=\mathbf{c}_1+\mathbf{c}_2+\mathbf{c}_3+\mathbf{c}_4\quad\mathrm{and}\quad f(\mathbf{e}_2)=\mathbf{c}_1-3\mathbf{c}_3+7\mathbf{c}_4. $$
V

Determine the mapping matrix for $f$ with respect to the basis $e$ in $\reel^2$ and basis $c$ in $\reel^4,.$

W

Solve the linear equation $\,f(\mx) =5\mc_1+3\mc_2-3\mc_3+17\mc_4\,.$