# Seatwork (Introduction to Linear Algebra)

Deadlines:

- Section B - 26 Feb 2024
- Section C - 26 Feb 2024

1. Draw the following vectors, in the Cartesian plane

$$
\vec{p}=\begin{bmatrix}
2\\3
\end{bmatrix},
\vec{q}=\begin{bmatrix}
-1\\2
\end{bmatrix},
\vec{r}=\begin{bmatrix}
-1\\-2
\end{bmatrix},
\vec{s}=\begin{bmatrix}
3\\-1
\end{bmatrix},
\vec{t}=\begin{bmatrix}
-2\\2
\end{bmatrix}
$$

2. (a) After applying a horizontal shear, draw the new location of the vectors $\vec{p}$,$\vec{q}$,$\vec{r}$,$\vec{s}$,$\vec{t}$. (b) After applying a vertical flip (flip with respect to the x-axis), draw the new location of the vectors $\vec{p}$,$\vec{q}$,$\vec{r}$,$\vec{s}$, and $\vec{t}$.

3. Apply the following transformation to the vectors above ($\vec{p}$,$\vec{q}$,$\vec{r}$,$\vec{s}$,$\vec{t}$). (No need to draw the transformed vectors)
$$
\begin{bmatrix}
-2&3\\
1&2
\end{bmatrix}
$$

4. (a) Find the composition of horizontal shear and vertical flip. (b) Find the composition of vertical flip and horizontal shear.

5. Find the determinant of the transformation in number (3).

6. Calculate the determinants of these transformations:

$$
A =\begin{bmatrix}
1&-1&-2\\
2&3&-4\\
-2&1&4
\end{bmatrix}, B =\begin{bmatrix}
2&-4&2\\
0&8&1\\
-3&2&5
\end{bmatrix}
$$

7. Find the inverses of the transformation matrices
$$
A =\begin{bmatrix}
1&-1&-2\\
2&3&-4\\
-2&1&4
\end{bmatrix}, B =\begin{bmatrix}
2&-4&2\\
0&8&1\\
-3&2&5
\end{bmatrix},
C=\begin{bmatrix}
3&2\\
-1&2
\end{bmatrix},
\\
D=\begin{bmatrix}
4&0.5\\
-2&-1
\end{bmatrix},
E=\begin{bmatrix}
-1.5&1&-0.5\\
4&-2&1\\
-0.5&0&-0.5
\end{bmatrix},
$$