Hugo Stokes
2022-10-18
Answered

Finding the inverse of linear transformation using matrix. A linear transformation represented by a matrix with respect to some random bases, how could I find the inverse of the transformation using the matrix representation?

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getrdone07tl

Answered 2022-10-19
Author has **23** answers

Let $T$ be an invertible linear transformation from an $n$-dimensional vector space to another, $A$ its $(n\times n)$ matrix, $\mathcal{B}=\{{\alpha}_{1},\dots ,{\alpha}_{n}\}$ a basis for the domain, ${\mathcal{B}}^{\prime}=\{{\beta}_{1},\dots ,{\beta}_{n}\}$ a basis for the co-domain, and $\beta ={y}_{1}{\beta}_{1}+\cdots +{y}_{n}{\beta}_{n}$ a vector in the co-domain.

The inverse of $T,{T}^{-1},$, is related to ${A}^{-1}$ by ${\left[{T}^{-1}\beta \right]}_{\mathcal{B}}={A}^{-1}[\beta {]}_{{\mathcal{B}}^{\prime}},$, where $[\cdot {]}_{\mathcal{B}}$ denotes the coordinate matrix relative to the ordered basis $\mathcal{B}.$ Then with ${A}_{ij}^{-1}$ representing the $ij$ entry of ${A}^{-1},$, we compute

$$\begin{array}{rl}{\left[{T}^{-1}\beta \right]}_{\mathcal{B}}& ={A}^{-1}[\beta {]}_{{\mathcal{B}}^{\prime}}\\ \\ & ={A}^{-1}[{y}_{1}{\beta}_{1}+\cdots +{y}_{n}{\beta}_{n}{]}_{{\mathcal{B}}^{\prime}}\\ \\ & ={A}^{-1}\left[\begin{array}{c}{y}_{1}\\ \vdots \\ {y}_{n}\end{array}\right]\\ \\ & =\left[\begin{array}{c}\sum _{i=1}^{n}{A}_{1i}^{-1}{y}_{i}\\ \vdots \\ \sum _{i=1}^{n}{A}_{ni}^{-1}{y}_{i}\end{array}\right];\\ \\ {T}^{-1}\beta & ={\alpha}_{1}\sum _{i=1}^{n}{A}_{1i}^{-1}{y}_{i}+\cdots +{\alpha}_{n}\sum _{i=1}^{n}{A}_{ni}^{-1}{y}_{i}.\end{array}$$

If you are satisfied having your inverse in terms of $\mathcal{B},$, we are done.

If, however, you want the inverse in terms of the standard ordered basis, express each ${\alpha}_{i}$ in terms of the standard basis, and group the terms in the last expression above by the elements of the standard basis.

The inverse of $T,{T}^{-1},$, is related to ${A}^{-1}$ by ${\left[{T}^{-1}\beta \right]}_{\mathcal{B}}={A}^{-1}[\beta {]}_{{\mathcal{B}}^{\prime}},$, where $[\cdot {]}_{\mathcal{B}}$ denotes the coordinate matrix relative to the ordered basis $\mathcal{B}.$ Then with ${A}_{ij}^{-1}$ representing the $ij$ entry of ${A}^{-1},$, we compute

$$\begin{array}{rl}{\left[{T}^{-1}\beta \right]}_{\mathcal{B}}& ={A}^{-1}[\beta {]}_{{\mathcal{B}}^{\prime}}\\ \\ & ={A}^{-1}[{y}_{1}{\beta}_{1}+\cdots +{y}_{n}{\beta}_{n}{]}_{{\mathcal{B}}^{\prime}}\\ \\ & ={A}^{-1}\left[\begin{array}{c}{y}_{1}\\ \vdots \\ {y}_{n}\end{array}\right]\\ \\ & =\left[\begin{array}{c}\sum _{i=1}^{n}{A}_{1i}^{-1}{y}_{i}\\ \vdots \\ \sum _{i=1}^{n}{A}_{ni}^{-1}{y}_{i}\end{array}\right];\\ \\ {T}^{-1}\beta & ={\alpha}_{1}\sum _{i=1}^{n}{A}_{1i}^{-1}{y}_{i}+\cdots +{\alpha}_{n}\sum _{i=1}^{n}{A}_{ni}^{-1}{y}_{i}.\end{array}$$

If you are satisfied having your inverse in terms of $\mathcal{B},$, we are done.

If, however, you want the inverse in terms of the standard ordered basis, express each ${\alpha}_{i}$ in terms of the standard basis, and group the terms in the last expression above by the elements of the standard basis.

asked 2021-09-18

I need to find a unique description of Nul A, namely by listing the vectors that measure the null space.

$A=\left[\begin{array}{ccccc}1& 5& -4& -3& 1\\ 0& 1& -2& 1& 0\\ 0& 0& 0& 0& 0\end{array}\right]$

asked 2021-09-13

Suppose that A is row equivalent to B. Find bases for the null space of A and the column space of A.

$A=\left[\begin{array}{ccccc}1& 2& -5& 11& -3\\ 2& 4& -5& 15& 2\\ 1& 2& 0& 4& 5\\ 3& 6& -5& 19& -2\end{array}\right]$

$B=\left[\begin{array}{ccccc}1& 2& 0& 4& 5\\ 0& 0& 5& -7& 8\\ 0& 0& 0& 0& -9\\ 0& 0& 0& 0& 0\end{array}\right]$

asked 2021-06-13

For the matrix A below, find a nonzero vector in the null space of A and a nonzero vector in the column space of A

$A=\left[\begin{array}{cccc}2& 3& 5& -9\\ -8& -9& -11& 21\\ 4& -3& -17& 27\end{array}\right]$

Find a vector in the null space of A that is not the zero vector

$A=\left[\begin{array}{c}-3\\ 2\\ 0\\ 1\end{array}\right]$

asked 2021-12-11

Detemine if $b$ is a linear combination of $a}_{1},{a}_{2},{a}_{3$

$${a}_{1}=\left[\begin{array}{c}2\\ 0\\ 2\end{array}\right],{a}_{2}=\left[\begin{array}{c}-4\\ 3\\ -4\end{array}\right],{a}_{3}=\left[\begin{array}{c}-5\\ 8\\ 4\end{array}\right],b=\left[\begin{array}{c}13\\ -4\\ 9\end{array}\right]$$

Choose the correct answer below

A. Vector$b$ is a linear combination of $a}_{1},{a}_{2},{a}_{3$ . The pivots in the corresponding echelon matrix are in the
first entry in the first column, the second entry in the second column, and the third entry in the third column.

B. Vector$b$ is not a lincar combination of $a}_{1},{a}_{2},{a}_{3$

C. Vector$b$ is a linear combination of $a}_{1},{a}_{2},{a}_{3$ . The pivots in the corresponding echelon matrix are in the
first entry in the first column and the third entry in the second column, and the third entry in the third column.

D. Vector$b$ is a linear combination of $a}_{1},{a}_{2},{a}_{3$ . The pivots in the corresponding echelon matrix are in the
first entry in the first column, the second entry in the second column, and the third entry in the fourth column.

Choose the correct answer below

A. Vector

B. Vector

C. Vector

D. Vector

asked 2020-11-20

Find the standard inner product on $P}^{2$ of the given polynomials $p=-5+4x+{x}^{2},q=3+4x\u20132{x}^{2}$ .

asked 2022-07-10

I'd like to be able to enter a vector or matrix, see it in 2-space or 3-space, enter a transformation vector or matrix, and see the result. For example, enter a 3x3 matrix, see the parallelepiped it represents, enter a rotation matrix, see the rotated parallelepiped.

asked 2022-05-20

Does there exist a matrix $A$ for which $AM$ = ${M}^{T}$ for every $M$. The answer to this is obviously no as I can vary the dimension of $M$. But now this lead me to think , if I take , lets say only $2\times 2$ matrix into consideration. Now for a matrix $M$, $A={M}^{T}{M}^{-1}$ so $A$ is not fixed and depends on $M$, but the operation follows all conditions of a linear transformation and I had read that any linear transformation can be represented as a matrix. So is the last statement wrong or my argument flawed?