Tagged: isomorphism of vector spaces

An Orthogonal Transformation from $\R^n$ to $\R^n$ is an Isomorphism

Problem 592

Let $\R^n$ be an inner product space with inner product $\langle \mathbf{x}, \mathbf{y}\rangle=\mathbf{x}^{\trans}\mathbf{y}$ for $\mathbf{x}, \mathbf{y}\in \R^n$.

A linear transformation $T:\R^n \to \R^n$ is called orthogonal transformation if for all $\mathbf{x}, \mathbf{y}\in \R^n$, it satisfies
\[\langle T(\mathbf{x}), T(\mathbf{y})\rangle=\langle\mathbf{x}, \mathbf{y} \rangle.\]

Prove that if $T:\R^n\to \R^n$ is an orthogonal transformation, then $T$ is an isomorphism.

 
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Find the Inverse Linear Transformation if the Linear Transformation is an Isomorphism

Problem 553

Let $T:\R^3 \to \R^3$ be the linear transformation defined by the formula
\[T\left(\, \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} \,\right)=\begin{bmatrix}
x_1+3x_2-2x_3 \\
2x_1+3x_2 \\
x_2+x_3
\end{bmatrix}.\]

Determine whether $T$ is an isomorphism and if so find the formula for the inverse linear transformation $T^{-1}$.

 
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Is the Linear Transformation Between the Vector Space of 2 by 2 Matrices an Isomorphism?

Problem 528

Let $V$ denote the vector space of all real $2\times 2$ matrices.
Suppose that the linear transformation from $V$ to $V$ is given as below.
\[T(A)=\begin{bmatrix}
2 & 3\\
5 & 7
\end{bmatrix}A-A\begin{bmatrix}
2 & 3\\
5 & 7
\end{bmatrix}.\] Prove or disprove that the linear transformation $T:V\to V$ is an isomorphism.

 
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Isomorphism of the Endomorphism and the Tensor Product of a Vector Space

Problem 80

Let $V$ be a finite dimensional vector space over a field $K$ and let $\End (V)$ be the vector space of linear transformations from $V$ to $V$.
Let $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n$ be a basis for $V$.
Show that the map $\phi:\End (V) \to V^{\oplus n}$ defined by $f\mapsto (f(\mathbf{v}_1), \dots, f(\mathbf{v}_n))$ is an isomorphism.
Here $V^{\oplus n}=V\oplus \dots \oplus V$, the direct sum of $n$ copies of $V$.
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