Jan 2, 2025 · This will create a system of simultaneous equations which can be solved to find the invariant point(s) If there is a line of invariant points, rather than solving to find a single solution (a point), the two equations will be able to simplify to the same equation. This means that there are infinitely many solutions, and therefore infinitely ...

Therefore, B(4,-1) is an invariant point (under the linear transformation M). The image of B is B itself. Note that the origin (0,0) is always an invariant point - this is a property of linear transformations. Lines of invariant points A Line of Invariant Points - A(2,3), O(0,0), and B(-1, − 3 2) are invariant, lying on the line 3 x = 2 y ...

E.g. 4 Find the invariant points under the transformation . Invariant lines A line is an invariant line under a transformation if the image of a point on the line is also on the line. It does not mean that every point on the line must map onto itself. N.B. Any …

An invariant point under a transformation is a point that maps to itself. As noted in part 4, linear transformations map the origin to the origin, so the origin is always an invariant point under a linear transformation. An invariant line is a line that maps to itself. To be precise, every point on the invariant line maps to a point on the line ...

Aug 27, 2018 · If so, they give a line of invariant points. So I tried this question using that method (and the method needed for finding invariant lines): For [this] matrix, find any lines of invariant points and any other invariant lines through the origin. $\begin{pmatrix}3&-2\\ 4&-3\end{pmatrix}$

Aug 22, 2024 · Link to Lesson: A-Level Further Mathematics Core Pure 1: https://mr-mathematics.com/product/further-mathematics-linear-transformation-invariant-points-and-in...

Every point on the line =−𝑥 4 is transformed to itself under the transformation @ 2 4 3 13 A. Also, every point on this line is transformed to the point @ 0 0 A under the transformation @ 1 4 3 12 A (which has a zero determinant). (5) Find the line of invariant points under the transformation given by the matrix @ 0 −1 −1 0 A @ 0 −1 ...

mirror line. Invariant lines An invariant line for a transformation is a line which is mapped to itself by the transformation. It is important to understand the difference between a line of invariant points and an invariant line. On a line of invariant points, all points are mapped to themselves. However, on an invariant line, all points are ...

Jun 3, 2024 · I know the difference; a line of invariant points is a line that includes all invariant points (points that map to themselves by matrix multiplication). Whereas, an invariant line is the same but any point on that line is mapped to any point on that line, meaning they don't necessarily map to themselves. I get how to work out invariant lines, but I don't get how to work out a line …

Aug 23, 2023 · Invariant Line: A line that remains the same or parallel after a given transformation where all points on the line are either invariant points or shifted to other points on the same line. Finding invariant points: Techniques vary depending on the transformation type, e.g., reflection (points lie on the line of reflection), rotation (centre of ...