Question

TRUE OR FALSE

If V is a subspace of ${\mathit{R}}^{\mathbf{n}}$and $\overrightarrow{\mathbf{x}}$is a vector space in ${\mathit{R}}^{\mathbf{n}}$, then the inequality $\overrightarrow{\mathbf{x}}\mathbf{·}\mathbf{\left(}\mathbf{pr}{\mathbf{o}}_{\mathbf{V}}\overrightarrow{\mathbf{x}}\mathbf{\right)}\mathbf{⩾}\mathbf{0}$must hold.

Short answer

The given statement is true.

Step-by-step solution

Definition of orthogonal projection

Vector $pro{j}_v\overrightarrow{x}$is orthogonal to vector $\overrightarrow{x}-pro{j}_v\overrightarrow{x}$.

Explanation of the statement

Consider the statement as follows.

“If V is a subspace of $R^n$and $\overrightarrow{x}$is a vector in $R^n$, then the inequality $\overrightarrow{x}·pro{j}_v\overrightarrow{x}⩾0$”.

Since,$projv\overrightarrow{x}$is a orthogonal vector$\overrightarrow{x}-pro{j}_v\overrightarrow{x}$and as follows.

$projv\overrightarrow{x}·\left(\overrightarrow{x}-pro{j}_v\overrightarrow{x}\right)=0$ ……(1)

$$\begin{gathered} \overrightarrow{x}·projv\overrightarrow{x}=\left(\overrightarrow{x}+projv\overrightarrow{x}-projv\overrightarrow{x}\right)·\left(projv\overrightarrow{x}\right) \\ =\left(projv\overrightarrow{x}+\left(\overrightarrow{x}+projv\overrightarrow{x}\right)\right)·\left(projv\overrightarrow{x}\right) \\ =\left(projv\overrightarrow{x}\right)·\left(projv\overrightarrow{x}\right)+\left(\overrightarrow{x}+projv\overrightarrow{x}\right)·\left(projv\overrightarrow{x}\right) \\ \overrightarrow{x}·projv\overrightarrow{x}=\left(projv\overrightarrow{x}\right)·\left(projv\overrightarrow{x}\right)+0 \end{gathered}$$

Simplify further as follows.

$$\begin{gathered} \overrightarrow{x}·projv\overrightarrow{x}=\left(projv\overrightarrow{x}\right)·\left(projv\overrightarrow{x}\right)+0 \\ ={||projv\overrightarrow{x}||}^2 \\ ⩾0 \end{gathered}$$

Thus, the given condition “If V is a subspace of$R^n$ and$\overrightarrow{x}$ is a vector in $R^n$, then the inequality $\overrightarrow{x}·pro{j}_v\overrightarrow{x}⩾0$” is true.