Question

Consider a linear transformation T from${\mathbf{ℝ}}^{\mathbf{n}}$to${\mathbf{ℝ}}^{\mathbf{p}}$and some linearly independent vectors${\overrightarrow{\mathbf{v}}}_{\mathbf{1}}\mathbf{,}{\overrightarrow{\mathbf{v}}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}{\overrightarrow{\mathbf{v}}}_{\mathbf{m}}$in${\mathbf{ℝ}}^{\mathbf{n}}$. Are the vectors$\mathit{T}\left({\overrightarrow{v}}_1\right)\mathbf{,}\mathit{T}\left({\overrightarrow{v}}_2\right)\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathit{T}\left({\overrightarrow{v}}_m\right)$necessarily linearly independent? How can ${\mathbf{ℝ}}^{\mathbf{p}}$you tell?

Short answer

If T is a linear transformation from ${\mathrm{ℝ}}^n$to${\mathrm{ℝ}}^p$ and some linearly independent vectors${\overrightarrow{v}}_1,{\overrightarrow{v}}_2,...,{\overrightarrow{v}}_m$ in ${\mathrm{ℝ}}^n$then the vectors$T\left({\overrightarrow{v}}_1\right)T,\left({\overrightarrow{v}}_1\right),..T\left({\overrightarrow{v}}_m\right)$ are necessarily linearly independent.

Step-by-step solution

  Mentioning the concept

Since it is given that the vectors ${\overrightarrow{v}}_1,{\overrightarrow{v}}_2,...{\overrightarrow{v}}_m$in ${\mathrm{ℝ}}^n$are linearly independent, then there exists scalars $a_1,a_2,...a_m$, all of them 0, such that

$a_1{\overrightarrow{v}}_1+a_2{\overrightarrow{v}}_2,...+a_m{\overrightarrow{v}}_m=0$

To prove the vectors T(v→1),T(v→2),...,T(v→m)are linearly independent

Since we have

$a_1{\overrightarrow{v}}_1+a_2{\overrightarrow{v}}_2+...+a_m{\overrightarrow{v}}_m=0$

Since it is given that T is a linear transformation from ${\mathrm{ℝ}}^n$to ${\mathrm{ℝ}}^p$, then we have

$$\begin{gathered} T\left(a_1{\overrightarrow{v}}_1+a_2{\overrightarrow{v}}_2+...+a_m{\overrightarrow{v}}_m\right)=T\left(0\right) \\ \Rightarrow T\left(a_1{\overrightarrow{v}}_1\right)+\left(a_2{\overrightarrow{v}}_2\right)+...+T\left(a_m{\overrightarrow{v}}_m\right)=\overrightarrow{0} \\ \Rightarrow a_{1}T\left({\overrightarrow{v}}_1\right)+a_{2}T\left({\overrightarrow{v}}_1\right)+...+a_{m}T\left({\overrightarrow{v}}_m\right)=\overrightarrow{0} \end{gathered}$$

But since the scalars$a_1,a_2,...,a_m$ are all zero, therefore the vector $T\left({\overrightarrow{v}}_1\right),T\left({\overrightarrow{v}}_2\right),...,T\left(\overrightarrow{v}m\right)$are linearly independent.