Basis and dimension
Exercises
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In $\mathbb{R}^5$ consider the following vector spaces $W: {2x_2 + x_4 - x_5 = 0 \text{ and } x_2 - x_3 = 0 \text{ and } 2x_2 - x_3 = 0}$ and $V = {(0,0,0,1,1), (-1,0,0,2,1), (1,0,0,2,3), (0,1,0,0,1)}$.
i) Compute the dimension and a basis of $W, V$
ii) Compute the dimension and a basis of $W \cap V \subseteq W + V$
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In $\mathbb{R}^5$ consider the following vector spaces $W = {(2,0,0,2,2), (-3,0,0,-1,-1), (0,0,2,-1,-1)}$ and $V = {(-1,2,1,0,0), (0,0,0,1,5), (0,1,1,-1,-3)}$.
i) Compute the dimension, a basis and the equations of $W, V$
ii) Compute the dimension, a basis and the equations of $W \cap V \subseteq W + V$
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In $\mathbb{R}^5$ consider the following vector spaces $W = {(1,1,1,1), (2,0,0,2,2), (-3,0,0,-1,-1), (0,1,0,0,1)}$ and $V = {(0,2,2,-6), (-1,2,1,0,0), (0,0,0,1,5), (0,1,1,-1,-3)}$.
i) Compute the dimension, a basis and the equations of $W, V$
ii) Compute the dimension, a basis and the equations of $W \cap V \subseteq W + V$