Question 3 · Álgebra UOC Juny 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?

$u_1 =$

$u_2 =$

$u_3 =$