Also, say if $T$ is one-one, does this mean it is a matrix transformation and hence a linear transformation?

aangenaamyj
2022-07-07
Answered

If $T:{\mathbb{R}}^{n}\to {\mathbb{R}}^{m}$ is a matrix transformation, does $T</math\; depend\; on\; the\; dimensions\; of$ \mathbb{R}$?\; i.e.,\; is$ T$one-one\; if$ m>n$,$ m=n$,\; or$ n>m$?$

Also, say if $T$ is one-one, does this mean it is a matrix transformation and hence a linear transformation?

Also, say if $T$ is one-one, does this mean it is a matrix transformation and hence a linear transformation?

You can still ask an expert for help

Monserrat Cole

Answered 2022-07-08
Author has **12** answers

Any matrix transformation $T:{\mathbb{R}}^{n}\to {\mathbb{R}}^{m}$ is a linear transformation (and vice versa once you've specified bases). If $n>m$, then $T$ cannot be injective because there cannot be $n$ linearly independent vectors in ${\mathbb{R}}^{m}$. Finally, if $n=m$ or $n<m$, one can say nothing about injectivity without more information. For instance, for $n=m$ you have the zero matrix transformation (not injective) and the identity matrix transformation.

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 2021-09-13

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 2021-09-18

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 2022-10-13

Let $A=4\times 4$ matrix: $\left[\begin{array}{cccc}3& 2& 10& -6\\ 1& 0& 2& -4\\ 0& 1& 2& 3\\ 1& 4& 10& 8\end{array}\right]$, let $b=4\times 1$ matrix: $\left[\begin{array}{c}-1\\ 3\\ -1\\ 4\end{array}\right]$

Is $b$ in the range of linear transformation $x\to Ax$?

Is $b$ in the range of linear transformation $x\to Ax$?

asked 2022-06-06

Let there be a linear transformation going from ${\mathbb{R}}^{3}$ to ${\mathbb{R}}^{2}$, defined by $T(x,y,z)=(x+y,2z-x)$. Find the transformation matrix if base 1:

$\u27e8(1,0,-1),(0,1,1),(1,0,0)\u27e9$,

base 2: $\u27e8(0,1),(1,1)\u27e9$

An attempt at a solution included calculating the transformation on each of the bases in ${\mathbb{R}}^{3}$, (base 1) and then these vectors, in their column form, combined, serve as the transformation matrix, given the fact they indeed span all of ${B}_{1}$ in ${B}_{2}$

Another point: if the basis for ${\mathbb{R}}^{3}$ and ${\mathbb{R}}^{2}$ are the standard basis for these spaces, the attempt at a solution is a correct answer.

$\u27e8(1,0,-1),(0,1,1),(1,0,0)\u27e9$,

base 2: $\u27e8(0,1),(1,1)\u27e9$

An attempt at a solution included calculating the transformation on each of the bases in ${\mathbb{R}}^{3}$, (base 1) and then these vectors, in their column form, combined, serve as the transformation matrix, given the fact they indeed span all of ${B}_{1}$ in ${B}_{2}$

Another point: if the basis for ${\mathbb{R}}^{3}$ and ${\mathbb{R}}^{2}$ are the standard basis for these spaces, the attempt at a solution is a correct answer.

asked 2022-01-31

1) Calculate the transformation matrix

2) Calculate the dimension of the kernel of the transformation, justify.

$T:{\mathbb{R}}^{\mathbb{3}}\to {\mathbb{R}}^{\mathbb{3}}$ is a linear transformation such that:

$T\left({\overrightarrow{e}}_{1}\right)={\overrightarrow{e}}_{1}-{\overrightarrow{e}}_{2}+2{\overrightarrow{e}}_{3}$

$T({\overrightarrow{e}}_{1}+{\overrightarrow{e}}_{2})=2{\overrightarrow{e}}_{3}$

$T({\overrightarrow{e}}_{1}+{\overrightarrow{e}}_{2}+{\overrightarrow{e}}_{3})=-{\overrightarrow{e}}_{2}+{\overrightarrow{e}}_{3}$

2) Calculate the dimension of the kernel of the transformation, justify.

asked 2021-11-23

If we have a matrices

$$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\text{and}{e}_{12}(\lambda )=\left[\begin{array}{cc}1& \lambda \\ 0& 1\end{array}\right]$$

then by doing product$$A{e}_{12}(\lambda )=\left[\begin{array}{cc}a& a\lambda +b\\ c& c\lambda +d\end{array}\right]\text{and}{e}_{12}(\lambda )A=\left[\begin{array}{cc}a+c\lambda & b+d\lambda \\ c& d\end{array}\right]$$

we can interpret that right multiplication by$e}_{12$ to A gives a column-operation: add $\lambda$ -times first column to the second column.
In similar way, left multiplication by ${e}_{12}\left(\lambda \right)$ to A gives row-operation on A.

Is there any conceptual (not computational, if any) way to see that elementary row and column operations on a matrix can be expressed as multiplication by elementary matrices on left or right, accordingly?

then by doing product

we can interpret that right multiplication by

Is there any conceptual (not computational, if any) way to see that elementary row and column operations on a matrix can be expressed as multiplication by elementary matrices on left or right, accordingly?