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 vector \( \mathbf{x}_{U} \) belongs to the subspace \( U \) and \( \mathbf{x}_{\perp U} \) 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 \[ \mathbf{x} = P_V\mathbf{x} + \mathbf{x}_{\perp V}. \]
FOR ERIC: This first Sketch is revised much. Since the vectors \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \) are fixed on the \( x_1 x_2 \)-plane, we can easily have a look from "above" on the right side of the Sketch.
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.
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?
FOR ERIC: in Sketches 2-4 we have "genuine" space vectors \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \). The right side figure describes the span of them, and \( \mathbf{v}_1 \) is fixed to point to the right, while \( \mathbf{v}_2 \) is drawn according the angle between them. The angle in space is somewhat funny, belonging to \( \left[0,\pi\right[ \). What do you think of this? In the Sketch 2 the viewpoint is fixed, while in 3-4 you may look from different directions (restrictedly, however). You look at the Sketch's left figure from direction of the vector \( S \) pointing to your eye from the origin.
FOR ERIC: in Sketches 2-4 we have "genuine" space vectors \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \). The right side figure describes the span of them, and \( \mathbf{v}_1 \) is fixed to point to the right, while \( \mathbf{v}_2 \) is drawn according the angle between them. The angle in space is somewhat funny, belonging to \( \left[0,\pi\right[ \). What do you think of this? In the Sketch 3 the viewpoint can be slightly adjusted with the coordinates \( s_1 \), \( s_1 \) and \( s_3 \) of the vector \( S \) pointing to your eye from the origin.
FOR ERIC: in Sketches 2-4 we have "genuine" space vectors \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \). The right side figure describes the span of them, and \( \mathbf{v}_1 \) is fixed to point to the right, while \( \mathbf{v}_2 \) is drawn according the angle between them. The angle in space is somewhat funny, belonging to \( \left[0,\pi\right[ \). What do you think of this? Here you don't see anymore the values of the vector \( \mathbf{v}_i \) coordinates, just roughly from the variable lines. You see \( P \). In the Sketch 4 the viewpoint can be slightly adjusted with the coordinates \( s_1 \), \( s_1 \) and \( s_3 \) of the vector \( S \) pointing to your eye from the origin.
a) Does the property observed in the preceding picture stay valid when you change parameters \( s_1 \), \( s_2 \) and/or \( s_3 \) ? yes no don't know b) How does the vector \( S = (s_1, s_2, s_3) \) look like on the screen ?
FOR ERIC: Sketches 5-6 remind of the first Sketch. Now we have fixed orthonormal bases. But the projection is not to a plane but a line. Perhaps we should make one more earlier for an arbitrary line? The projection done using two-vector-span with Gramm matrix and projection matrix does not work for parallel vectors. But we can do the same process for one vector and line, n'est-ce pas?
a) Compute the matrix \( P_S \) of the orthogonal projection on the subspace spanned by \( S \).
b) Compute the matrix \( P_{\perp S} \) of the orthogonal projection on the plane orthogonal to \( S \).
c) Let \( \mathbf{i} \), \( \mathbf{j} \), \( \mathbf{k} \) be an orthonormal basis of the Euclidean 3-space and let \( \mathbf{S} = s_1 \mathbf{i} + s_2 \mathbf{j} + s_3 \mathbf{k} \). Compute the lengths of the projections on the orthogonal complement of \( \mathbf{S} \), of the vectors \( \mathbf{i} \), \( \mathbf{j} \), \( \mathbf{k} \), \( \mathbf{j} {-} \mathbf{k} \), \( \mathbf{k} {-} \mathbf{i} \), \( \mathbf{i} {-} \mathbf{j} \).
d) Do your results fit with the pictures 2 and 3? yes no don't know