Question

TRUE OR FALSE

26. If Vis a subspace of ${\mathit{R}}^{\mathbf{n}}$and $\overrightarrow{\mathbf{x}}$is a vector in ${\mathit{R}}^{\mathbf{n}}$, then vector $\mathit{p}\mathit{r}\mathit{o}{\mathbf{j}}_{\mathbf{V}}\overrightarrow{\mathbf{x}}$must be orthogonal to vector $\overrightarrow{\mathbf{x}}\mathbf{-}\mathit{p}\mathit{r}\mathit{o}{\mathbf{j}}_{\mathbf{V}}\overrightarrow{\mathbf{x}}$.

Short answer

The given statement is true.

Step-by-step solution

Step-by-step solution Step 1: Determine the definition of an orthogonal matrix

A square matrix with real numbers or elements is said to be an orthogonal matrix if its transpose is equal to its inverse matrix.

Or it can be said that, when the product of a square matrix and its transpose gives an identity matrix, then the square matrix is known as an orthogonal matrix.

Consider an example

For example,

Suppose A is a square matrix with real elements and of $n\times n$order and $A^T$ is the transpose of A. Then according to the definition, if, $A^T=A^{-1}$is satisfied, then,

$A·A^T=I$

Because the $pro{j}_V\overrightarrow{x}$ component is taken out of vector x.

Thus, vector $pro{j}_V\overrightarrow{x}$ must be orthogonal to vector $\overrightarrow{x}-pro{j}_V\overrightarrow{x}$.

Hence, the statement is true.