Question

Find image and kernel of the linear transformation$\mathit{L}\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{(}\mathit{A}\mathbf{+}{\mathit{A}}^{\mathbf{T}}\mathbf{)}$from ${\mathbf{ℝ}}^{\mathbf{n}\mathbf{\times }\mathbf{n}}\mathbf{}\mathit{t}\mathit{o}\mathbf{}{\mathbf{ℝ}}^{\mathbf{n}\mathbf{\times }\mathbf{n}}$.

Short answer

The image of L is symmetric matrices and kernel of L is skew-symmetric matrices.

Step-by-step solution

Determine the image of T

Consider the linear transformation $L\left(A\right)=\frac{1}{2}(A+A^T)$from ${\mathrm{ℝ}}^{n\times m}\mathrm{to}{\mathrm{ℝ}}^{n\times m}$.

Assume$A\in {\mathrm{ℝ}}^{n\times n}$such that $L\left(B\right)=A^T$where $A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& ...& a_{1n}\\ a_{21}& a_{22}& ...& a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{n1}& a_{n2}& ...& a_{33}\end{array}\right]$and role="math" localid="1660119537916" $\left[\begin{array}{cccc}{b}_{11}& b_{12}& ...& b_{1n}\\ b_{21}& b_{22}& ...& b_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ b_{n1}& b_{n2}& ...& b_{33}\end{array}\right]$.

Substitute the value

$$\begin{gathered} B^T={\left[\begin{array}{cccc}{b}_{11}& b_{12}& ...& b_{1n}\\ b_{21}& b_{22}& ...& b_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ b_{n1}& b_{n2}& ...& b_{33}\end{array}\right]}^T \\ {B}^T=\left[\begin{array}{cccc}{b}_{11}& b_{12}& ...& b_{1n}\\ b_{21}& b_{22}& ...& b_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ b_{n1}& b_{n2}& ...& b_{33}\end{array}\right] \end{gathered}$$

Substitute the values $\left[\begin{array}{cccc}{b}_{11}& b_{12}& ...& b_{1n}\\ b_{21}& b_{22}& ...& b_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ b_{n1}& b_{n2}& ...& b_{33}\end{array}\right]$for $B^T$and $\left[\begin{array}{cccc}{b}_{11}& b_{12}& ...& b_{1n}\\ b_{21}& b_{22}& ...& b_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ b_{n1}& b_{n2}& ...& b_{33}\end{array}\right]$for B in the equation $A=\frac{1}{2}(B+B^T)$as follows.

$$\begin{gathered} A=\frac{1}{2}(B+B^T) \\ =\frac{1}{2}\left(\left[\begin{array}{cccc}{b}_{11}& b_{12}& ...& b_{1n}\\ b_{21}& b_{22}& ...& b_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ b_{n1}& b_{n2}& ...& b_{33}\end{array}\right]+\left[\begin{array}{cccc}{b}_{11}& b_{12}& ...& b_{1n}\\ b_{21}& b_{22}& ...& b_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ b_{n1}& b_{n2}& ...& b_{33}\end{array}\right]\right) \\ =\left[\begin{array}{cccc}2b_{11}& b_{12}+b_{21}& ...& b_{1n}+b_{1n}\\ b_{21}+b_{12}& 2b_{22}& ...& b_{2n}+b_{n2}\\ ⋮& ⋮& \ddots & ⋮\\ b_{n1}+b_{1n}& b_{n2}+b_{2n}& ...& 2b_{33}\end{array}\right] \\ \left[\begin{array}{cccc}{b}_{11}& b_{12}& ...& b_{1n}\\ b_{21}& b_{22}& ...& b_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ b_{n1}& b_{n2}& ...& b_{33}\end{array}\right]=\left[\begin{array}{cccc}{b}_{11}& (b_{12}+b_{21})/2& ...& b_{1n}\\ (b_{21}+b_{12})/2& b_{22}& ...& (b_{2n}+b_{n2})/2\\ ⋮& ⋮& \ddots & ⋮\\ (b_{n1}+b_{1n})/2& (b_{n2}+b_{2n})/2& ...& b_{33}\end{array}\right] \end{gathered}$$

Compare the equation both side as follows.

$$\begin{gathered} a_{jj}=b_{jj} \\ {a}_{ij}=a_{ji}(b_{ij}+b_{ji})/2 \end{gathered}$$

Therefore, the image of the linear transformation is the set of all symmetric matrix.

Determine the kernel of

Assume $A\in ker\left(L\right)$, by the definition L(A) = 0.

Compare the equations $L\left(A\right)=0\mathrm{and}L\left(A\right)=\frac{1}{2}(A+A^T)$as follows.

role="math" localid="1660120645510" $\frac{1}{2}(A+A^T)=0$

Simplify the equation $\frac{1}{2}(A+A^T)=0$as follows.

$$\begin{gathered} \frac{1}{2}(A+A^T)=0 \\ A+A^T=0 \\ A=-A^T \end{gathered}$$

Therefore, the kernel of linear transformation L is set of all $n\times n$skew symmetric matrix.

Hence, the image of L is symmetric matrices and kernel of L is skew-symmetric matrices.