Examen de Matemàtiques URV 2024 · Exemples

Problem 3. Sum and intersection of vector subspaces

Consider the vector subspaces $U$ and $V$ of $\mathbb{R}^4$, where $a \in \mathbb{R}$ is a real parameter:$$U = \{(\beta, \alpha - \beta + \gamma, \gamma, a\alpha) \mid \alpha, \beta, \gamma \in \mathbb{R}\}$$$$V = \{(x, y, z, t) \in \mathbb{R}^4 \mid -x - 2y + t = 0,\ az - t = 0\}$$Determine, as a function of the parameter $a$, a basis and the dimension for the subspaces $U + V$ and $U \cap V$.

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Problem 2. Eigenvalues, eigenvectors and matrix power

Consider the matrix$$A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 2 & 0 \\ 1 & 0 & 1 \end{pmatrix}$$

a)

Find the characteristic polynomial, the eigenvalues, and the corresponding eigenspaces of $A$.

b)

Compute $A^n$ for any positive integer $n \ge 1$.

Problem 1. Endomorphism bijectivity, kernel and image

Let $f_d: \mathbb{R}^3 \to \mathbb{R}^3$ be the endomorphism defined by:$$f_d(x, y, z) = (x + 5y + dz,\ x + dy - z,\ x + 5y)$$

a)

Determine for which values of the real parameter $d$ the endomorphism $f_d$ is bijective.

b)

Find the value of $d$ for which the vector $(5, -1, 5)$ belongs to $\ker(f_d)$.

c)

Find the value of $d$ for which the vector $(-5, 1, 0)$ belongs to $\ker(f_d) \cap \operatorname{Img}(f_d)$.