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I want to find the basis of the following subspace

$$W = \{(x_1, x_2, x_3, x_4)^T \in \mathbb{R}^4 : x_1 + x_2 + x_3 + x_4 = 1\}$$

We clearly see that, from the constraint, we get the condition

$$x_4 = 1 - x_1 - x_2 - x_3$$

so these are the vectors of the form

$$(x_1, x_2, x_3, 1-x_1-x_2-x_3)^T$$

and this space should have dimension $3$, but how do I proceed to find it explicitly?

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  • $\begingroup$ Hint: start with $\vec {v_1}=(1,0,0,-1)\in W$. Can you find a vector in $W$ which is linearly independent from that? And so on. $\endgroup$ – lulu Dec 30 '19 at 13:50
  • $\begingroup$ @lulu, your $\vec v_1\not\in W$ $\endgroup$ – Martund Dec 30 '19 at 13:51
  • $\begingroup$ $W$ is not a subspace of $\mathbb{R^{4}}.$ $\endgroup$ – math Dec 30 '19 at 13:51
  • $\begingroup$ @Martund you are correct! I read the condition as requiring that $\sum x_i=0$. As stated, $W$ isn't a subspace. $\endgroup$ – lulu Dec 30 '19 at 13:52
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As already remarked in the comments, $W$ is not a linear subspace of $R^4$. What you can do is the following: the condition $$x_4=1−x_1−x_2−x_3$$ allows you to find a parametrization for $W$. By choosing the first three $x_i$ as free variables, e.g. $$x_1 = r, x_2 = s, x_3 = t$$ you can write $$ (x_1,x_2,x_3,x_4) = (r,s,t,1-r-s-t) = (0,0,0,1) + r (1,0,0,-1) + s (0,1,0,-1) + t (0,0,1,-1).$$ Your set $W$ is the subspace spanned by the latter three vectors translated over $(0,0,0,1)$.

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