Find equations of lines passing through the origin which are invariant under the transformation represented by A.
Understand the Problem
The question is asking to find equations of lines that pass through a specific point and are invariant under the transformation represented by the matrix A. This involves understanding the properties of linear transformations and the geometric interpretation of invariance.
Answer
The invariant lines are of the form $y = mx$ where $m$ satisfies the invariance condition derived from the transformation.
Answer for screen readers
The equations of the invariant lines passing through the origin is given by $$ y = mx $$ where $m$ satisfies the equation derived from invariance conditions.
Steps to Solve
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Identify the transformation matrix We have the matrix $$ A = \begin{pmatrix} 2 & 3 \ 0 & 5 \end{pmatrix}. $$
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Determine the invariant lines A line in vector form can be represented as $$ \mathbf{y} = m \mathbf{x} + \mathbf{b} $$ where $\mathbf{b}$ is a point on the line. To be invariant under transformation by $A$, the transformation of points on the line must also lie on the same line.
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Formulate the equation for line invariance The transformed point is given by $$ A \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 2x + 3y \ 5y \end{pmatrix}. $$ For a line to be invariant, the relation must hold for all points $(x, y)$ on the line.
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Set up the relation for invariance To find the relationship for the slope $m$ of the line, we substitute $y = mx + b$ into the transformed equation: $$ A \begin{pmatrix} x \ mx + b \end{pmatrix} = \begin{pmatrix} 2x + 3(mx + b) \ 5(mx + b) \end{pmatrix}. $$ This results in the points $(x, mx + b)$ transforming into $(2x + 3(mx + b), 5(mx + b))$.
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Solve for invariance conditions We need to find $m$ such that for any constant $b$, the slope remains the same post-transformation. Specifically, we will equate the slopes obtained from the transformed coordinates.
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Extract the conditions Equating slopes from the transformed coordinates gives us: $$ \frac{5(mx + b)}{2x + 3(mx + b)} = m. $$ Simplifying this expression will yield the invariant slope.
The equations of the invariant lines passing through the origin is given by $$ y = mx $$ where $m$ satisfies the equation derived from invariance conditions.
More Information
Invariant lines under a linear transformation represented by a matrix maintain their slope post-transformation. This concept often arises in linear algebra, particularly concerning eigenvalues and eigenvectors.
Tips
- Not applying the transformation correctly to both coordinates of the point.
- Forgetting to consider that both sides of the equation must yield the same slope for all $x$.
- Miscalculating the simplified expression after equating slopes.
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