Examen de Álgebra UOC 2023 · Juny

Question 1 (25% of final grade). Complex numbers

Solve the following items involving complex numbers:

a.

Determine the complex number in algebraic (binomial) form $\dfrac{x^2}{\overline{y}\cdot z}$, taking $x = 2 + i$, $y = -3 + 2i$, and $z = -i$.

Question 1 (25% of final grade). Complex numbers

Solve the following items involving complex numbers:

b.

Calculate all fourth roots of the given complex number and write the solutions in polar representation $r_\theta$ with the arguments $\theta \in [0, 2\pi)$:

$$\sqrt[4]{81\left(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\right)}$$

$\theta_0 =$

$\theta_1 =$

$\theta_2 =$

$\theta_3 =$

Question 2 (25% of final grade). Linear systems with parameters

Consider the parameter $a \in \{0, 1, \dots, 9\}$ and the following system of linear equations in three variables $x, y, z$:$$\begin{cases}x + y + z = 1 \\ (a + 1)x + ky + (a + 1)z = a + 1 \\ (a + 1)x + (a + 1)y + kz = a + 1 \\ (a + 1)x + (a + 1)y + (a + 1)z = k\end{cases}$$

a.

Classify the system according to the values of the real parameter $k \in \mathbb{R}$.

b.

Discard the fourth equation to obtain a square system of three equations in three unknowns. Find its general solution for the particular case $k = 2023$.

Question 3 (25% of final grade). Subspaces and change of basis

Let $E \subset \mathbb{R}^4$ be the 3-dimensional subspace defined by:$$E = \{(a_1, a_2, a_3, a_4) \in \mathbb{R}^4 \mid a_1 + a_2 + a_3 = a_4\}$$Let $v = (a+1, a+2, a+3, 3a+6)$, where $a \in \{0, 1, \dots, 9\}$.

a)

Verify that the set $A = \{(1,0,0,1), (0,1,0,1), (0,0,1,1)\}$ forms a basis of $E$. Check whether $v \in E$, and if so, determine the coordinates of $v$ relative to the basis $A$.

$c_1 =$

$c_2 =$

$c_3 =$

b)

Let $k \in \mathbb{R}$, $k \neq 0$, and let the change-of-basis matrix from a basis $B = \{u_1, u_2, u_3\}$ to the basis $A$ be:$$C_{B\to A} = \begin{pmatrix} k & -\frac{\sqrt{6}}{6} & -\frac{\sqrt{3}}{6} \\ 0 & \frac{\sqrt{6}}{3} & -\frac{\sqrt{3}}{6} \\ 0 & 0 & \frac{\sqrt{3}}{2} \end{pmatrix}$$Find the vectors of the basis $B$. Is there any value of $k$ for which $B$ is an orthonormal basis of $E$ with respect to the standard Euclidean inner product?

$u_1 =$

$u_2 =$

$u_3 =$

Question 4 (25% of final grade). Affine transformations in the plane

Let $a \in \{0, 1, \dots, 9\}$ be a parameter.

a)

Construct the affine transformation matrix in homogeneous coordinates for a homothety (scaling) with ratio $r = a + 2$ centered at the point $P = (-5, a)$. Find the image of the point $Q = (1, 1)$ under this transformation.

$x' =$

$y' =$

b)

The following matrix in homogeneous coordinates defines a rotation in $\mathbb{R}^2$:$$M = \begin{pmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} & (2-\sqrt{3})(a+1) \\ \frac{1}{2} & \frac{\sqrt{3}}{2} & -(a+1) \\ 0 & 0 & 1 \end{pmatrix}$$Find the angle of rotation $\alpha$ and the coordinates of the center of rotation.

$x_C =$

$y_C =$