Chapter 5.CR, Problem 42CR

Repeat Exercise $41$ for $B=\left\{\left(-1,2,2\right),\left(1,0,0\right)\right\}$ and $x=\left(-3,4,4\right)$.

Let $B=\left\{\left(0,2,-2\right),\left(1,0,-2\right)\right\}$ be a basis for a subspace of $R^3$, and consider $x=\left(-1,4,-2\right)$, a vector in the subspace.

(a) Write $x$ as a linear combination of the vectors in $B$.That is, find the coordinates of $x$ relative to $B$.

(b) Apply the Gram-Schmidt orthonormalization process to transform $B$ into an orthonormal set $B^{\prime }$.

(c) Write $x$ as a linear combination of the vectors in $B^{\prime }$.That is, find the coordinates of $x$ relative to $B^{\prime }$.