Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 31E from Chapter 4.3 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 31E
Chapter:
Problem:
Show that if is linearly dependent in V, then the set of images, , is linearly dependent in W . This fact shows that if a linear transformation maps a set onto a linearly independent set then the original set is linearly independent, too (because it cannot be linearly dependent).
Step-by-Step Solution
Given Information
We have to show that if the set $\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}, \ldots ,{{\bf{v}}_p}} \right\}$ is linearly dependent in V, then the set $ \left\{ T \left( \mathbf { v } _ { 1 } \right) , T \left( \mathbf { v } _ { 2 } \right) , T \left( \mathbf { v } _ { 3 } \right) , \ldots , T \left( \mathbf { v } _ { p } \right) \right\} $ is linearly dependent in W for the given transformation. \[ T : V \rightarrow W \]
Step 1:
If the set of vectors $\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}, \ldots ,{{\bf{v}}_p}} \right\}$ is linearly independent in V then for nonzero scalars, the following relation must be satisfied. \[ c _ { 1 } \mathbf { v } _ { 1 } + c _ { 2 } \mathbf { v } _ { 2 } + \ldots + c _ { p } \mathbf { v } _ { p } = 0 \]
Step 2:
Apply linear transformation to above equation \[ \begin{aligned} T \left( c _ { 1 } \mathbf { v } _ { 1 } + c _ { 2 } \mathbf { v } _ { 2 } + \ldots + c _ { p } \mathbf { v } _ { p } \right) & = T ( 0 ) \\ c _ { 1 } T \left( \mathbf { v } _ { 1 } \right) + c _ { 2 } T \left( \mathbf { v } _ { 2 } \right) + \ldots + c _ { p } T \left( \mathbf { v } _ { p } \right) & = 0 \end{aligned} \] Since the scalars are not all zero, the set $\left\{ {T\left( {{{\bf{v}}_1}} \right),T\left( {{{\bf{v}}_2}} \right),T\left( {{{\bf{v}}_3}} \right), \ldots ,T\left( {{{\bf{v}}_p}} \right)} \right\}$ is linearly independent in W.
The statement is true that if the set $\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}, \ldots ,{{\bf{v}}_p}} \right\}$ is linearly dependent in V, then the set $ \left\{ T \left( \mathbf { v } _ { 1 } \right) , T \left( \mathbf { v } _ { 2 } \right) , T \left( \mathbf { v } _ { 3 } \right) , \ldots , T \left( \mathbf { v } _ { p } \right) \right\} $ is linearly dependent in W
We have to show that if the set $\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}, \ldots ,{{\bf{v}}_p}} \right\}$ is linearly dependent in V, then the set $ \left\{ T \left( \mathbf { v } _ { 1 } \right) , T \left( \mathbf { v } _ { 2 } \right) , T \left( \mathbf { v } _ { 3 } \right) , \ldots , T \left( \mathbf { v } _ { p } \right) \right\} $ is linearly dependent in W for the given transformation. \[ T : V \rightarrow W \]
Step 1:
If the set of vectors $\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}, \ldots ,{{\bf{v}}_p}} \right\}$ is linearly independent in V then for nonzero scalars, the following relation must be satisfied. \[ c _ { 1 } \mathbf { v } _ { 1 } + c _ { 2 } \mathbf { v } _ { 2 } + \ldots + c _ { p } \mathbf { v } _ { p } = 0 \]
Step 2:
Apply linear transformation to above equation \[ \begin{aligned} T \left( c _ { 1 } \mathbf { v } _ { 1 } + c _ { 2 } \mathbf { v } _ { 2 } + \ldots + c _ { p } \mathbf { v } _ { p } \right) & = T ( 0 ) \\ c _ { 1 } T \left( \mathbf { v } _ { 1 } \right) + c _ { 2 } T \left( \mathbf { v } _ { 2 } \right) + \ldots + c _ { p } T \left( \mathbf { v } _ { p } \right) & = 0 \end{aligned} \] Since the scalars are not all zero, the set $\left\{ {T\left( {{{\bf{v}}_1}} \right),T\left( {{{\bf{v}}_2}} \right),T\left( {{{\bf{v}}_3}} \right), \ldots ,T\left( {{{\bf{v}}_p}} \right)} \right\}$ is linearly independent in W.
The statement is true that if the set $\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}, \ldots ,{{\bf{v}}_p}} \right\}$ is linearly dependent in V, then the set $ \left\{ T \left( \mathbf { v } _ { 1 } \right) , T \left( \mathbf { v } _ { 2 } \right) , T \left( \mathbf { v } _ { 3 } \right) , \ldots , T \left( \mathbf { v } _ { p } \right) \right\} $ is linearly dependent in W