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. \((a,b,c,d)^\mathsf T\to \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right)\).