Ejercicio Problem 3. Sum and intersection of vector subspac… · Matemàtiques URV Exemples 2024

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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