aangenaamyj

2022-07-07

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?

Monserrat Cole

Beginner2022-07-08Added 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.

An object moving in the xy-plane is acted on by a conservative force described by the potential energy function

where$U(x,y)=\alpha (\frac{1}{{x}^{2}}+\frac{1}{{y}^{2}})$ is a positive constant. Derivative an expression for the force expressed terms of the unit vectors$\alpha$ and$\overrightarrow{i}$ .$\overrightarrow{j}$ I need to find a unique description of Nul A, namely by listing the vectors that measure the null space

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$A=\left[\begin{array}{ccccc}1& 5& -4& -3& 1\\ 0& 1& -2& 1& 0\\ 0& 0& 0& 0& 0\end{array}\right]$T must be a linear transformation, we assume. Can u find the T standard matrix.$T:{\mathbb{R}}^{2}\to {\mathbb{R}}^{4},T\left({e}_{1}\right)=(3,1,3,1)\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}T\left({e}_{2}\right)=(-5,2,0,0),\text{}where\text{}{e}_{1}=(1,0)\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}{e}_{2}=(0,1)$

?Find a nonzero vector orthogonal to the plane through the points P, Q, and R. and area of the triangle PQR

Consider the points below

P(1,0,1) , Q(-2,1,4) , R(7,2,7).

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