Chapter 6 - Orthogonality and Least Squares - 6.4 Exercises - Page 361: 22

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Let c and d be scalars and $\vec{u}$ and $\vec{v}$ be vectors and U be the matrix whose columns are $u_1$ through $u_p$ after they have been normalized. Then, $T(c\vec{u}+d\vec{v})=proj_W(c\vec{u}+d\vec{v})=UU^T(c\vec{u}+d\vec{v})=UU^T(c\vec{u})+UU^T(d\vec{v})=cUU^T(\vec{u})+dUU^T(\vec{v})=c\ proj_W\vec{u}+d\ proj_W\vec{v}=cT(\vec{u})+dT(\vec{v})$