Find all invariant subspaces W ⊂ℝ^2 of the following linear transformations L: ℝ^3 →ℝ^3 : (a) th

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Find all invariant subspaces W ⊂ℝ^2 of the following linear...

Find all invariant subspaces $W \subset \mathbb{R}^2$ of the following linear transformations $L: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ : (a) the scaling transformation $(2 x, 3 y, 4 z)^T$; (b) the shear $(x+3 y, y, z)^T$; (c) counterclockwise rotation by a $45^{\circ}$ angle around the $x$-axis.

Find all invariant subspaces $W \subset \mathbb{R}^2$ of the following linear transformations $L: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ :
(a) the scaling transformation $(2 x, 3 y, 4 z)^T$;
(b) the shear $(x+3 y, y, z)^T$;
(c) counterclockwise rotation by a $45^{\circ}$ angle around the $x$-axis.

Key Concepts

Invariant Subspaces

An invariant subspace under a linear transformation is a subspace that is mapped into itself by the transformation. This idea is central to understanding and decomposing linear operators, as it allows one to study the behavior of the operator on simpler, lower-dimensional pieces of the space.

Diagonalization

Diagonalization is the process of finding a basis consisting entirely of eigenvectors, so that the linear transformation can be represented as a diagonal matrix. This method simplifies the study of invariant subspaces, as each basis vector (and the line it spans) is invariant under the transformation.

Shear Transformations and Jordan Forms

Shear transformations often are not diagonalizable and may lead to a Jordan block structure. In these cases, the analysis of invariant subspaces requires understanding not just eigenvectors but generalized eigenvectors, which together describe the subspace behavior when the operator cannot be fully diagonalized.

Rotational Transformations

Rotational transformations, such as rotations about an axis in a Euclidean space, have invariant subspaces that include the rotation axis itself. Additionally, any plane perpendicular to the axis, while not fixed pointwise, is transformed in a structured way, which is fundamental in understanding the geometric properties of rotations.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are elements that satisfy the equation T(v) = ?v for a given linear transformation. The one-dimensional subspaces spanned by eigenvectors are naturally invariant, and these components reveal important aspects of the structure and action of the operator on the space.

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