Question

Let {v1, v2, v3} be a set of nonzero vectors in R^m such that tr(vi)vj = 0 when i does not equal j. Show that the set is linearly independent.

Answer 1

\[v _{i}^{T}v_{j} = 0 \] when
\[i \neq j\]

Sorry if my notation for this stuff is bad, I'm new to it is all.

Answer 2

let \(\alpha,\beta,\gamma\) such that
\[\large \alpha v_1+\beta v_2+\gamma v_3=0. \]
Then
\[\large v_1^t(\alpha v_1+\beta v_2+\gamma v_3)=v_1^t0=0 \]
\[\large v_1^t(\alpha v_1)+v_1^t(\beta v_2)+v_1^t(\gamma)v_3=0 \]
\[\large \alpha(v_1^tv_1)+\beta(v_1^tv_2)+\gamma(v_1^tv_3)=0 \]
since all are mutualy orthogonal
\[\large \alpha(v_1^tv_1)=0 \]
since \(v_1\neq0\) then \(\alpha=0\).

The same can be dome for \(v_2\) and \(v_3\). So the \(v_i\) are linearly independent.