Introduction

We shall call dynamic figures briefly "Sketches". Usually big coloured points are movable (draggable) by computer mouse, while small points are fixed or move passively.

In the following Sketches we have two column vectors \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \) in \( \mathbb{R}^3 \). We have augmented (or catenated) them to the \( 3\times 2 \) matrix \[ V = (\mathbf{v}_1 \ \mathbf{v}_2) = (v_{ij}). \] Note, that boldface symbols like \( \mathbf{x} \), \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \) mean column vectors: \[ \mathbf{x} = \left( \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right), \quad \mathbf{v}_1 = \left( \begin{array}{c} v_{11} \\ v_{21} \\ v_{31} \end{array} \right), \quad \mathbf{v}_2 = \left( \begin{array}{c} v_{12} \\ v_{22} \\ v_{32} \end{array} \right). \] The aim is to project any vector \( \mathbf{x} \in \mathbb{R}^3 \) orthogonally to the subspace generated by the set \( \{\mathbf{v}_1, \mathbf{v}_2 \} \). This subspace is also called the span of \( \{\mathbf{v}_1, \mathbf{v}_2 \} \) and denoted by \( [\mathbf{v}_1, \mathbf{v}_2] \) or \( \span(\mathbf{v}_1, \mathbf{v}_2) \). We denote the orthogonal projection of \( \mathbf{x} \) briefly by \( \mathbf{x}' \) or \( \mathbf{x}_{V} \), if needed, instead of \( \mathbf{x}_{[\mathbf{v}_1, \mathbf{v}_2]} \).

Recall that in orthogonal projection to a subspace \( U \) the vector \( \mathbf{x} \) is decomposed in two mutually orthogonal components \[ \mathbf{x} = \mathbf{x}_{U} + \mathbf{x}_{\perp U}, \] where the projection vector \( \mathbf{x}_{U} \) belongs to the subspace \( U \) and the orthogonal component \( \mathbf{x}_{\perp U} \) is orthogonal to \( U \), which means that it is orthogonal to every vector of \( U \).
This generalizes the procedure of orthogonal projection done directly using the inner product, when the set \( \{\mathbf{v}_1, \mathbf{v}_2 \} \) is orthonormal. We can build the projection matrix \( P = P_V \) to do the job for any linearly independent vector set \( \{\mathbf{v}_1, \mathbf{v}_2 \} \).

Assume the vector set \( \{\mathbf{v}_1, \mathbf{v}_2 \} \) is linearly independent and \( V = (\mathbf{v}_1 \ \mathbf{v}_2) \).
Then the square matrix \( G := V^T V \), the Gramm matrix of \( V \), is regular and the inverse \( G^{-1} = (V^T V)^{-1} \) exists. The projection matrix is \[ P = P_V = V (V^T V)^{-1} V^T = V G^{-1} V^T, \] so that the orthogonal decomposition becomes \[ \mathbf{x} = P_V\mathbf{x} + \mathbf{x}_{\perp V}. \]

Sketch for Questions 1

In this Sketch you see four different "worlds":
1. The 3-space with the variable vector \( \mathbf{x} \) and the vectors \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \) generating the subspace \( [\mathbf{v}_1, \mathbf{v}_2] \).
2. The vector coordinate real lines for changing \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \).
3. The subspace of the 3-space, generated by the vectors \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \).
4. The size of the orthogonal component vector from \( \mathbf{x}' \) to \( \mathbf{x} \).

You may freely change the vector \( \mathbf{x} \) by its coordinates in world 1, while the vectors \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \) only restrictedly in world 2; using the coloured big points.

Questions 1

a) Why is vector \( \mathbf{x}' \mathbf{x} \) fixed when you move \( v_{11} \), \( v_{12} \), \( v_{21} \) or \( v_{22} \), when you see both of them?