The subspace V in $R^4$ is the span of vectors $u = \frac 1{3} [1, -1, 1, 2]^T$ and $v = \frac 1 {5}[1, 2, 1, 1]^T$. Find the shortest distance from the point $P(1, 1, 1, 1)$ and the subspace.
Don't really know where to start here so would really appreciate some hints and tips. I've looked up similar questions like this: Distance from point $(1,1,1,1)$ to the subspace of $R^4$ but don't really get the idea of it.
Best Answer
The shortest distance between a point to a subspace is equal to the distance between the point to its projection.
To find the projection, we can use the Gram-Schmidt process.
Let $w=(1,1,1,1)^T$.
Let $u_1 = \frac{u}{\|u\|}$, and $v_1 = \frac{v-(v^Tu_1)u_1}{\left\|v-(v^Tu_1)u_1 \right\|}$, then $u_1$ and $v_1$ forms an orthonormal basis.
and its projection is $r=(w^Tu_1)u_1+(w^Tv_1)v_1$ and the distance is $\|w-r\|$.