(a) Show that every invertible linear function L: ℝ^n →ℝ^n can be represented by the identity ma

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(a) Show that every invertible linear function L: ℝ^n →ℝ^n...

(a) Show that every invertible linear function $L: \mathbb{R}^n \rightarrow \mathbb{R}^n$ can be represented by the identity matrix by choosing appropriate (and not necessarily the same) bases on the domain and codomain. (b) Which linear transformations are represented by the identity matrix when the domain and codomain are required to have the same basis? (c) Find bases of $\mathbb{R}^2$ so that the following linear transformations are represented by the identity matrix: (i) the scaling map $S[\mathbf{x}]=2 \mathrm{x} ;$ (ii) counterclockwise rotation by $45^{\circ} ;$ (iii) the shear $\left(\begin{array}{rr}1 & 0 \\ -2 & 1\end{array}\right)$.

Key Concepts

Representation by the Identity Matrix

A matrix representation by the identity matrix means that, under the chosen bases, the transformation acts as if it simply maps each vector to itself. For an invertible transformation, by appropriately selecting distinct bases in the domain and codomain, the transformation can be 'normalized' to the identity matrix. When the same basis is used for both spaces, only the trivial transformation (the identity transformation) yields an identity matrix representation.

Similarity Transformations

A similarity transformation is the process by which a matrix is conjugated by an invertible matrix, effectively changing its basis representation. This concept is critical when determining canonical forms of matrices, as it shows that many properties of linear transformations, such as eigenvalues and rank, remain invariant under a change of basis. It underpins the idea that the intrinsic behavior of a linear map does not depend on the particular coordinate system used.

Change of Basis

Changing the basis in a vector space alters the matrix representing a linear transformation, even though the transformation itself remains unchanged. By selecting appropriate bases for the domain and codomain, one can simplify the form of the transformation’s matrix, sometimes even transforming it into the identity matrix or another canonical form. This process highlights the idea that the matrix representation is coordinate-dependent.

Invertible Linear Transformations

An invertible linear transformation is a bijective map between vector spaces that preserves the operations of vector addition and scalar multiplication. Its invertibility guarantees the existence of an inverse transformation, ensuring a one-to-one correspondence between elements of the domain and codomain. This is a foundational concept in linear algebra, emphasizing that the structure of the vector spaces is maintained under the transformation.

Matrix Representation of Linear Transformations

Every linear transformation between finite-dimensional vector spaces can be represented as a matrix once bases for the domain and codomain are chosen. This representation allows abstract properties of linear maps to be studied through concrete matrix operations, enabling computational techniques and theoretical insights, such as determining invertibility, eigenvalues, and the structure of the transformation.

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