Collection of Maths Problems

Isomorphic spaces

Task number: 4528

• Variant

  $\mathbb R^{2\times 2}$ and $\mathbb R^4$

• Solution

  The space $\mathbb R^{2\times 2}$ has dimension 4, the same as the space $\mathbb R^4$. They are both over the same field $\mathbb R$, and are therefore isomorphic to each other.

  Isomorphism can be determined by a bijective mapping of a basis to a basis, e.g. $f\left(\begin{smallmatrix}1 & 0 \end{smallmatrix}\right)=(1{,}0,0{,}0)^\mathsf T$, $f\left(\begin{smallmatrix}0 & 1 \\0 & 0 \end{smallmatrix}\right)=(0{,}1,0{,}0)^\mathsf T$, $f\left(\begin{smallmatrix}0 & 0 \\1 & 0 \end{smallmatrix}\right)=(0{,}0,1{,}0)^\mathsf T$, $f\left(\begin{smallmatrix}0 & 0 \\0 & 1 \end{smallmatrix}\right)=(0{,}0,0{,}1)^\mathsf T$.

  This isomorphism is then given by the formula: $f\left(\begin{smallmatrix}a & b \\c & d \end{smallmatrix}\right)=(a,b,c,d)^\mathsf T$.

• Variant

  $\mathbb R^4$ and the space of real polynomials of degree at most three

• Answer

  They are isomorphic, e.g. $(a,b,c,d)^\mathsf T\to ax^3+bx^2 +cx+d$.

• Variant

  $\mathbb R^{m\times n}$ and $\mathbb R^{n\times m}$

• Answer

  They are isomorphic, e.g. $\boldsymbol A \to \boldsymbol A^\mathsf T$.

• Variant

  $\mathbb R^n$ over $\mathbb R$ and $\mathbb C^n$ over $\mathbb C$

• Answer

  Even though they have the same dimension, they are not isomorphic because they are spaces above different fields.

• Variant

  $\mathbb R^2$ and $\{\boldsymbol x = (x_1, x_2, x_3, x_4)^\mathsf T \in \mathbb R^4\colon x_1 + x_2 = x_3 + x_4 = 0\}$

• Answer

  They are isomorphic, e.g. $(a,b)^\mathsf T\to (a,-a,b,-b)^\mathsf T$.

• Variant

  $\mathbb R^4$ and the space of linear mappings $f : \mathbb R^4 \to \mathbb R$

• Answer

  Each mapping $f$ is uniquely determined by its matrix, e.g. with respect to the standard bases $[f]_{E,E}$.

  The spaces are isomorphic, e.g. $(a,b,c,d)^\mathsf T\to (a\ b\ c\ d)$.

• Variant

  $\mathbb R^4$ and the space of linear mappings $f : \mathbb R \to \mathbb R^4$

• Answer

  They are isomorphic, e.g. $(a,b,c,d)^\mathsf T\to \left(\begin{smallmatrix}a \\ b \\ c \\ d \end{smallmatrix}\right)$.

• Variant

  $\mathbb R^4$ and the space of linear mappings $f : \mathbb R^2 \to \mathbb R^2$

• Answer

  They are isomorphic, e.g. $(a,b,c,d)^\mathsf T\to \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right)$.