Let $V$ be an inner product space over $\mathbb R.$ Let $u_1, u_2, u_3, u_4 \in V$ such that $\langle u_i,u_j\rangle < 0,$ for $i,j = 1, 2, 3, 4 i\neq j.$
Show that $u_1, u_2, u_3$ are linearly independent.
Please advise the ways to prove the result.
Example: $V=\mathbb{R}^4$, $u_1=(1,0,0,0)$, $u_2=(-1,2,0,0)$, $u_3=(-1,-1,3,0)$, and $u_4=(-1,-1,-1,1)$. This are all linearly independent because they generate $\mathbb{R}^4$. However, the inner products between each two are negative.
PS: the inner product is zero iff the vectors are orthogonal (that is the definition, but think of it geometrically). Also, a finite dimensional vector space has always a basis, and it has also an inner product, but that basis does not have to be an orthogonal basis. The example I gave was cooked so that the inner products are all negative, but imagine that they are all positive: $\{(1,0,0,0), (1,1,0,0), (1,1,1,0), (1,1,1,1)\}$ is a linearly independent set of vectors, and non is orthogonal to another.