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Chapter 1: Q31E (page 1)

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

Expert verified

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

01

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.

02

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\({c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} + {c_3}{{\bf{v}}_3} = 0\).

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

03

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.

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Most popular questions from this chapter

Suppose \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2}} \right\}\) is a linearly independent set in \({\mathbb{R}^n}\). Show that \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _1} + {{\mathop{\rm v}\nolimits} _2}} \right\}\) is also linearly independent.

Find the general solutions of the systems whose augmented matrices are given

11. \(\left[ {\begin{array}{*{20}{c}}3&{ - 4}&2&0\\{ - 9}&{12}&{ - 6}&0\\{ - 6}&8&{ - 4}&0\end{array}} \right]\).

A Givens rotation is a linear transformation from \({\mathbb{R}^{\bf{n}}}\) to \({\mathbb{R}^{\bf{n}}}\) used in computer programs to create a zero entry in a vector (usually a column of matrix). The standard matrix of a given rotations in \({\mathbb{R}^{\bf{2}}}\) has the form

\(\left( {\begin{aligned}{*{20}{c}}a&{ - b}\\b&a\end{aligned}} \right)\), \({a^2} + {b^2} = 1\)

Find \(a\) and \(b\) such that \(\left( {\begin{aligned}{*{20}{c}}4\\3\end{aligned}} \right)\) is rotated into \(\left( {\begin{aligned}{*{20}{c}}5\\0\end{aligned}} \right)\).

Suppose an experiment leads to the following system of equations:

\(\begin{aligned}{c}{\bf{4}}.{\bf{5}}{x_{\bf{1}}} + {\bf{3}}.{\bf{1}}{x_{\bf{2}}} = {\bf{19}}.{\bf{249}}\\1.6{x_{\bf{1}}} + 1.1{x_{\bf{2}}} = 6.843\end{aligned}\) (3)

  1. Solve system (3), and then solve system (4), below, in which the data on the right have been rounded to two decimal places. In each case, find the exactsolution.

\(\begin{aligned}{c}{\bf{4}}.{\bf{5}}{x_{\bf{1}}} + {\bf{3}}.{\bf{1}}{x_{\bf{2}}} = {\bf{19}}.{\bf{25}}\\1.6{x_{\bf{1}}} + 1.1{x_{\bf{2}}} = 6.8{\bf{4}}\end{aligned}\) (4)

  1. The entries in (4) differ from those in (3) by less than .05%. Find the percentage error when using the solution of (4) as an approximation for the solution of (3).
  1. Use your matrix program to produce the condition number of the coefficient matrix in (3).

Determine whether the statements that follow are true or false, and justify your answer.

15: The systemAx鈬赌=[0001]isinconsistent for all 4脳3 matrices A.
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