Exercise 2.B.7

Question

Prove or give a counterexample: If $v_1, v_2, v_3, v_4$ is a basis of $V$ and $U$
is a subspace of $V$ such that $v_1, v_2 \in U$ and $v_3 
otin U$ and $v_4 
otin U$, then
$v_1, v_2$ is a basis of U.

Arden Shi · Accepted Answer

Let 
$$
V=R^4
$$ 
and 
$$
U = \{(x_1, x_2, x_3, 0) \in R^4 \}
$$
As you can see, $U$ is a subspace of $V$.
Let 
$$
v_1, v_2, v_3, v_4 = (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 1), (0, 0, 0, 1)
$$
$v_1, v_2, v_3, v_4$ is clearly a basis of $V$, $v_1, v_2 \in U$, and $v_3, v_4 
otin U$

However, $v_1, v_2$ is not a basis of $U$.
Therefore, it is disproved.

Exercise 2.B.7

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Exercise 2.B.7

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