Question

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) be a linear transformation, and let \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}} \right\}\) be a linearly dependent set in \({\mathbb{R}^n}\). Explain why the set \(\left\{ {T\left( {{{\bf{v}}_1}} \right),T\left( {{{\bf{v}}_2}} \right),T\left( {{{\bf{v}}_3}} \right)} \right\}\) is linearly dependent.

Short answer

The set \(\left\{ {T\left( {{{\bf{v}}_1}} \right),T\left( {{{\bf{v}}_2}} \right),T\left( {{{\bf{v}}_3}} \right)} \right\}\)is linearly dependent.

Step-by-step solution

Write the condition for the set to be linearly dependent

The vectors are said to be linearly dependent if the equation is in the form of \({c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} + ... + {c_p}{{\bf{v}}_p} = 0\), where \({c_1},{c_2},...,{c_p}\) are scalars.

Check whether the set \(\left\{ {T\left( {{{\bf{v}}_1}} \right),T\left( {{{\bf{v}}_2}} \right),T\left( {{{\bf{v}}_3}} \right)} \right\}\) is linearly dependent or not

It is given that \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}} \right\}\)is a linearly dependent set in \({\mathbb{R}^n}\).

To prove\(\left\{ {T\left( {{{\bf{v}}_1}} \right),T\left( {{{\bf{v}}_2}} \right),T\left( {{{\bf{v}}_3}} \right)} \right\}\)is linearly dependent, take linear transformation on both sides of the equation\({c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} + {c_3}{{\bf{v}}_3} = 0\).

\(\begin{array}{c}T\left( {{c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} + {c_3}{{\bf{v}}_3}} \right) = T\left( 0 \right)\\T\left( {{c_1}{{\bf{v}}_1}} \right) + T\left( {{c_2}{{\bf{v}}_2}} \right) + T\left( {{c_3}{{\bf{v}}_3}} \right) = T\left( 0 \right)\\{c_1}T\left( {{{\bf{v}}_1}} \right) + {c_2}T\left( {{{\bf{v}}_2}} \right) + {c_3}T\left( {{{\bf{v}}_3}} \right) = T\left( 0 \right)\end{array}\)

Give the reason for the set \(\left\{ {T\left( {{{\bf{v}}_1}} \right),T\left( {{{\bf{v}}_2}} \right),T\left( {{{\bf{v}}_3}} \right)} \right\}\) that is linearly dependent

The obtained equation\({c_1}T\left( {{{\bf{v}}_1}} \right) + {c_2}T\left( {{{\bf{v}}_2}} \right) + {c_3}T\left( {{{\bf{v}}_3}} \right) = T\left( 0 \right)\)shows that the vectors are linearly dependent.

Thus, the set \(\left\{ {T\left( {{{\bf{v}}_1}} \right),T\left( {{{\bf{v}}_2}} \right),T\left( {{{\bf{v}}_3}} \right)} \right\}\)is linearly dependent.