$\left[\begin{array}{cc}-2& 11\\ 4& 2\end{array}\right]$ represents a linear transformation T:${\mathbb{R}}^{2}$ to ${\mathbb{R}}^{2}$ with respect to the basis, [(3,1),(0,2)]. Find the matrix of T with respect to basis [(1,1),(−1,1)]

phepafalowl
2022-07-20
Answered

$\left[\begin{array}{cc}-2& 11\\ 4& 2\end{array}\right]$ represents a linear transformation T:${\mathbb{R}}^{2}$ to ${\mathbb{R}}^{2}$ with respect to the basis, [(3,1),(0,2)]. Find the matrix of T with respect to basis [(1,1),(−1,1)]

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I would like to ask how to prove these two formulas.

for T is a scalar and r is radial component of a vector in spherical coordinate.

(1) $\mathbf{r}\cdot \mathbf{[}\mathbf{(}\mathbf{r}\cdot \mathrm{\nabla}\mathbf{)}\mathrm{\nabla}T+2\mathrm{\nabla}T]=\mathbf{r}\cdot \mathrm{\nabla}T+\mathbf{r}\cdot \mathrm{\nabla}\mathbf{(}\mathbf{r}\cdot \mathrm{\nabla}T)$

(2) ${\mathrm{\nabla}}^{2}(T\mathbf{r}\mathbf{)}=\mathbf{r}{\mathrm{\nabla}}^{2}T+T{\mathrm{\nabla}}^{2}\mathbf{r}+2\mathrm{\nabla}T\cdot \mathrm{\nabla}\mathbf{r}$

for T is a scalar and r is radial component of a vector in spherical coordinate.

(1) $\mathbf{r}\cdot \mathbf{[}\mathbf{(}\mathbf{r}\cdot \mathrm{\nabla}\mathbf{)}\mathrm{\nabla}T+2\mathrm{\nabla}T]=\mathbf{r}\cdot \mathrm{\nabla}T+\mathbf{r}\cdot \mathrm{\nabla}\mathbf{(}\mathbf{r}\cdot \mathrm{\nabla}T)$

(2) ${\mathrm{\nabla}}^{2}(T\mathbf{r}\mathbf{)}=\mathbf{r}{\mathrm{\nabla}}^{2}T+T{\mathrm{\nabla}}^{2}\mathbf{r}+2\mathrm{\nabla}T\cdot \mathrm{\nabla}\mathbf{r}$

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Finding $\mathrm{ker}[A{]}^{T}$

Let

$A=\left(\begin{array}{cccc}1& 1& -1& 2\\ 1& 1& 0& 3\\ -1& 0& 1& 0\end{array}\right).$

I have to calculate $\mathrm{ker}[A]$ and $\mathrm{ker}[A{]}^{T}$

I proved that $\mathrm{dim}(\mathrm{I}\mathrm{m}[A])=3$ with basis

$\left[\begin{array}{c}1\\ 1\\ -1\end{array}\right],\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right],\left[\begin{array}{c}-1\\ 0\\ 1\end{array}\right]$

and $\mathrm{dim}(\mathrm{ker}[A])=1$ with basis

$\left[\begin{array}{c}-1\\ -2\\ -1\\ 1\end{array}\right],$

but i have problems with $\mathrm{ker}[A{]}^{T}$. Do i have to calculate ${A}^{T}$ and then, with that matrix, finding the ker?

Let

$A=\left(\begin{array}{cccc}1& 1& -1& 2\\ 1& 1& 0& 3\\ -1& 0& 1& 0\end{array}\right).$

I have to calculate $\mathrm{ker}[A]$ and $\mathrm{ker}[A{]}^{T}$

I proved that $\mathrm{dim}(\mathrm{I}\mathrm{m}[A])=3$ with basis

$\left[\begin{array}{c}1\\ 1\\ -1\end{array}\right],\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right],\left[\begin{array}{c}-1\\ 0\\ 1\end{array}\right]$

and $\mathrm{dim}(\mathrm{ker}[A])=1$ with basis

$\left[\begin{array}{c}-1\\ -2\\ -1\\ 1\end{array}\right],$

but i have problems with $\mathrm{ker}[A{]}^{T}$. Do i have to calculate ${A}^{T}$ and then, with that matrix, finding the ker?