Geometrically: This is exactly the equation for a plane through $\,(0,0,0)\,$ with $\,\mv\,$ as the normal vector. Find the solution in standard parametric form, then you have the two basis vectors that span the complement.
answer
A basis is $\,(\,(-2,1,0),(-3,0,1)\,)\,.$
C
Find in $\,\reel^3\,$ a basis for the orthogonal complement to the subspace spanned by $\,(1,1,0)\,$ and $\,(0,2,1)\,.$
answer
A basis is the cross product $\,(1,-1,2)\,.$
D
Find in $\,\reel^4\,$ a basis for the orthogonal complement to the subspace spanned by $\,(1,-1,2,5)\,$ and $\,(0,1,0,-2)\,.$
hint
A homogeneous system of linear equations - consisting of two equations in four unknowns - emerges. Find the solution in standard parametric form, then you have two basis vectors that spans the complement.
answer
A basis is $\,(\,(-2,0,1,0),(-3,2,0,1)\,)\,.$
Exercise 2: When There is $\,n\,$ Different Eigenvalues
E
Why is it particular easy to diagonalize a symmetric $\,n\times n\,$-matrix by orthogonal substitution, if it has $\,n\,$different eigenvalues?
answer
If you take an eigenvector belonging to each of the eigenvalues then the collected set of eigenvectors is already an orthogonal basis for $\,\reel^n\,.$ If you norm the vectors (so they all have the length 1), you have an orthonormal basis for $\,\reel^n\,.$ The matrix that appears when you put the basis vectors together, can diagonalize the given symmetric matrix (by orthogonal substitution).
A $\,3\times 3$-matrix $\,\mA\,$ has been treated in Maple, like this:
F
State $\,\mA\,$ and explain that it is symmetric.
G
Let $\,f\,$ denote the linear map that has the mapping matrix $\,\mA\,$ with respect to the ordinary basis in $\,\reel^3\,.$ Determine an orthonormal basis for $\,\reel^3\,$ consisting of eigenvectors for $\,f\,,$ and state the mapping matrix that represents $\,f\,$ with respect to the orthonormal basis found.
hint
You norm a vector by dividing the vector with its length. Consider e.g. $\,(1,1,1)\,.$ Its length is $\,\sqrt{1^2+1^2+1^2}=\sqrt{3}\,.$ Therefore the normed vector is $\,(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})\,.$
Enter the basis vectors from the previous answer as columns in a $\,3\times 3$-matrix, then you have $\,Q\,.$ The corresponding diagonal matrix $\,\mathbf{\Lambda}\,$ is the one that was given in the previous answer.
Exercise 3: Eigenspaces with gm > 1
A $\,3\times 3$-matrix $\,\mB\,$ has been treated in Maple, like this:
I
State $\,\mB\,$ and explain that it is symmetric.
J
Determine an orthogonal matrix $\,Q\,$ and a diagonal matrix $\,\mathbf{\Lambda}\,$ such that
%$\,\mathbf Q\transp\cdot\mB\cdot\mathbf Q=\mathbf{\Lambda}\,.$
Maple gives a set of three linearly independent eigenvectors. But the two that belong to $\,E_{10}\,,$ are obviously not othrgonal (their dot product is not 0). Use Gram-Schmidt to find an orthonormal basis $\,(\,\mathbf q_1,\mathbf q_2)\,$ for $\,E_{10}\,.$ Then $\,\mathbf q_3=\mathbf q_1\times \mathbf q_2\,$ lies in $\,E_{1}\,,$ the orthogonal complement to $\,E_{10}\,.$ Therefore $\,(\,\mathbf q_1,\mathbf q_2,\mathbf q_3)\,$ is an orthonormal basis for $\,\reel^3\,$ consisting of eigenvectors for $\,\mB\,.$
Exactly eight possible orthonormal bases for $\,\reel^2\,$ consisting of eigenvectors for $\,f\,$ exists. Make a drawing where the basis vectors drawn from the origin are shown.
But the first basis vector does not have to lie in the 1. quadrant…
L
Four of the eight bases have the ordinary orientation. Show that the orthogonal matrix that belongs to each of the four is positive orthogonal, while the other four are negative orthogonal.
answer
One of the four bases that has the ordinary orientation, that is given in the previous answer. The determinant of its corresponding orthogonal matrix is 1. Therefore it is positive orthogonal.
Exercise 5: Positive Orthogonal Matrix as Mapping Matrix
Every positive orthogonal matrix in $\,\reel^{2\times 2}\,$ can be written in the form
Note that $\,u\,$ is the direction angle for the first basis vector $\,(\cos(u),\sin(u))\,.$ Or more precisely: The q-coordinate-system appears as a rotation of the standard coordinate system through the angle $\,u\,.$ Now we shall investigate how $\,\mQ\,$ functions as a mapping matrix!
A
Explain that the image $\,\my=\mathbf Q\cdot\mx\,$ appears after rotating the vector $\,\mx\,$ through the angle $\,u\,,$ see the figure
hint
It is essential to show that the coordinate vector for $\,\my\,$ with respect to the new basis is identical to the coordinate vector for $\,\mx\,$ in the standard base.
$$\vekind qy=\matind qMe\cdot\vekind ey
=\matind qMe\cdot(\matind eMq\cdot\vekind ex)
=(\matind qMe\cdot\matind eMq)\cdot\vekind ex
=\vekind ex\,.$$
QED.
B
Open the GeoGebra-sheet OrthogonalMap . Verify that while $\,\mQ\,$ maps $\,\mx\,$ in $\,\my\,$ by rotation through the angle $\,u\,,$ then $\,\mQ\transp\,$ does the opposit: maps $\,\mx\,$ in $\,\mz\,$ by rotation through the angle $\,-u\,.$
$\,(\,\mathbf q_1,\mathbf q_2)\,$ is a basis corresponding to $\,\mQ\,$ while $\,(\,\mathbf q_3,\mathbf q_4)\,$ is a basis corresponding to $\,\mQ\transp\,$.
$\,\mQ\,$ maps $\,\mx\,$ in $\,\my\,$ and $\,\mQ\transp\,$ maps $\,\mx\,$ in $\,\mz\,$.
Move the vector $\,\mx\,$. What happens to $\,\mx\,$, $\,\my\,$ and $\,\mz\,$ regarding lengths and angles?
Move the vector $\,\mathbf q_1\,$. What happens to $\,\mx\,$, $\,\my\,$ and $\,\mz\,$ regarding lengths and angles?
Exercise 6: Analysis of a Symmetric Map
Assume that a symmetric $\,2\times 2\,$ matrix $\,\mA\,$ has been diagonalized by a positive orthogonal substitution, like this:
In the equation $\,\mathbf Q\transp\cdot\mA\cdot\mathbf Q=\mathbf{\Lambda}\,$ you multiply on both sides by $\,\mQ\,$ from the left and $\,\mQ\transp\,$ from the right.
D
In continuation of this: Explain that therefore a symmetric map is composed like this:
First the object is rotated through the angle -$u\,$ where $\,u\,$ denotes the directional angle for the first column in $\,\mQ\,.$
The rotated object is scaled by the factor $\,\lambda_1\,$ in the direction of the first axis and by a factor $\,\lambda_2\,$ in the direction of the second axis.
The scaled object it rotated (back wards!) by the angle $\,u\,.$
as a mapping matrix for geometric vectors in the plane drawn from the origin. Find an angle of rotation $\,u\,$ that enters into step 1 and 3 of the map. And determine the factors that in step 2 are needed for scaling in the direction of the first axis and second axis, respectively.
Now we shall factorize and anlyze the mapping matrix
