Let $u$ and $v$ be non-collinear vectors in $R^2$. Let $w$ be the orthogonal projection vector of $u$ on v.…
Let $u$ and $v$ be non-collinear vectors in $R^2$. Let $w$ be the orthogonal projection vector of $u$ on
v. Consider two statements :
(i) Any vector in $R^2$ can be written as a linear combination of $u$ and $v$
(ii) $w$ can be written as a linear combination of $u$ and $v$ as $w=a u+b v$, where both $a$ and $b$ are non-zero real numbers.
Both (i) and (ii) are true
Only (i) is true, but (ii) is false
Only (ii) is true, but (i) is false
Both (i) and (ii) are false
Solution
Given $\mathbf{u}$ and $\mathbf{v}$ be non-collinear vectors in $R^2$
and $\mathbf{w}$ be the orthogonal projection vector of $\mathbf{u}$ on $\mathbf{v}$.
Any vector in $R^2$ can be written as linear combination of $\mathbf{u}$ and $\mathbf{v}$ because $u \neq \lambda v$ for any constant $\lambda$.
Hence, $\{\mathbf{u}, \mathbf{v}\}$ set is linearly independent but $\mathbf{w}$ can not be written as $a \mathbf{u}+b \mathbf{v}$ because $\mathbf{w} . \mathbf{u} \neq 0$ and $\mathbf{W} . \mathbf{v} \neq 0$, which is contradiction of orthogonal projection concept.