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First published on Thursday, Jan 9, 2025 and last modified on Thursday, Jan 9, 2025
Mathedu
Basis, Transition Matrix
In that test, you will discover the Gram-Schmidt Process to build an orthonormal basis with any basis of the plane.
And you will use that orthonormal basis to calculate the coordinates of some vectors in the original basis without any matrix inversion.
Consider for the rest of the test the ordered set \( B=(\overrightarrow{I},\overrightarrow{J})\) , with column vectors of coordinates in the canonical basis \( B_0=(\overrightarrow{i},\overrightarrow{j})\) :
\( [\overrightarrow{I}]_{B_0}=\begin{bmatrix} 1\\ 2 \end{bmatrix}\) and \( [\overrightarrow{J}]_{B_0}=\begin{bmatrix} -2\\ 2 \end{bmatrix}\) .
Determine the matrix \( P\) with columns the \( [\overrightarrow{I}]_{B_0}\) and \( [\overrightarrow{J}]_{B_0}\) and calculate its determinant.
Deduce that \( B=(\overrightarrow{I},\overrightarrow{J})\) is a basis of the plane.
Calculate the inverse of \( P\) . Leave \( \frac{1}{\det(P)}\) in factor.
Deduce the coordinates in \( B\) of the following vectors:
\( \overrightarrow{u}_1\) such that \( [\overrightarrow{u}_1]_{B_0}=\begin{bmatrix} 1\\ 1 \end{bmatrix}\) .
\( \overrightarrow{u}_2\) such that \( [\overrightarrow{u}_2]_{B_0}=\begin{bmatrix} -3\\ 1 \end{bmatrix}\) .
\( \overrightarrow{u}_3\) such that \( [\overrightarrow{u}_1]_{B_0}=\begin{bmatrix} -1\\ -2 \end{bmatrix}\) .
Consider the basis \( B'=(\overrightarrow{I'},\overrightarrow{J'})\) such that:
\( \overrightarrow{I'}=\overrightarrow{I}\) , and \( \overrightarrow{J'}=\overrightarrow{J}-\frac{\overrightarrow{I}\cdot\overrightarrow{J}} {\overrightarrow{I}\cdot\overrightarrow{I}}\overrightarrow{I}\) .
Without using the numerical values, prove that \( \overrightarrow{I'}\bot\overrightarrow{J'}\) .
Calculate the numerical values of \( [\overrightarrow{I'}]_{B_0}\) and \( [\overrightarrow{J'}]_{B_0}\) .
Set \( \frac{6}{5}\) in factor in \( [\overrightarrow{J'}]_{B_0}\) .
Check that the vectors of the \( B'\) are orthogonal.
Now that we have two orthogonal vectors, we derive an orthonormal basis while normalizing them.
The basis is \( B''=(\overrightarrow{I''},\overrightarrow{J''})\) such that \( \overrightarrow{I''}=\frac{\overrightarrow{I'}}{\left\|\overrightarrow{I'}\right\|}\) and \( \overrightarrow{J''}=\frac{\overrightarrow{J'}}{\left\|\overrightarrow{J'}\right\|}\) .
Calculate the norms of \( \overrightarrow{I'}\) and \( \overrightarrow{J'}\) , and \( [\overrightarrow{I''}]_{B_0}\) and \( [\overrightarrow{J''}]_{B_0}\) .
Put \( \sqrt{5}\) to the numerator and set \( \frac{\sqrt{5}}{5}\) in factor for both vectors.
Check that the basis \( B''\) is orthonormal.
Calculate the vectors of the basis \( B''\) as a function of the vectors of the basis \( B\) . Use their definitions in the questions 3 and 4.
Deduce the transition matrix \( Q\) from the basis \( B''\) to the basis \( B\) .
Set \( \frac{\sqrt{5}}{5}\) in factor.
Calculate the coordinates of the three vectors of the question 2 in the orthonormal basis \( B''\) . Just use dot products and leave \( \frac{\sqrt{5}}{5}\) in factor.
Deduce the coordinates of these vectors in the basis \( B\) and check for each vector that they are the same as in the question 2.
The Gram-Schmidt orthonormalisation process may be generalized to any dimension, where the inversion of matrices are not at all as trivial as in dimension 2.
And it allows to calculate the coordina†es of any vector in any basis with just dot products and matrix multiplication.
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