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Chapter 1: Q31Q (page 1)

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) be a linear transformation, with A its standard matrix. Complete the following statement to make it true: “T is one-to-one if and only if A has ­­____ pivot columns.” Explain why the statement is true. [Hint:Look in the exercises for Section 1.7.]

Short Answer

Expert verified

T is one-to-one if and only if A has ­­\(n\) pivot columns.

Step by step solution

01

Identify the condition for the transformation of dimensions

For matrix Aof order\(m \times n\), if the vector\({\bf{x}}\)is in\({\mathbb{R}^n}\), then transformation\(T\)of vector x is represented as\(T\left( x \right)\), and it is in the dimension\({\mathbb{R}^m}\).

It can also be written as\(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\).

Here, the dimension \({\mathbb{R}^n}\) is the domain, and the dimension \({\mathbb{R}^m}\) is the codomain of transformation \(T\).

02

Complete the statement to make it true

If the columns of A are linearly independent, then transformation T is one-to-one.

Matrix A has n pivot columns if they are linearly independent.

Thus, the correct statement is “T is one-to-one if and only if A has ­­\(n\) pivot columns.”

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Most popular questions from this chapter


Consider two vectors v→1 andv→2in R3 that are not parallel.

Which vectors inlocalid="1668167992227" ℝ3are linear combinations ofv→1andv→2? Describe the set of these vectors geometrically. Include a sketch in your answer.

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on an \(xy\)-graph: u, v, \( - {\bf{v}}\), \( - 2{\bf{v}}\), u + v , u - v, and u - 2v. Notice that u - vis the vertex of a parallelogram whose other vertices are u, 0, and \( - {\bf{v}}\).

4. u and v as in Exercise 2

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 elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.

29. \(\left[ {\begin{array}{*{20}{c}}0&{ - 2}&5\\1&4&{ - 7}\\3&{ - 1}&6\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}1&4&{ - 7}\\0&{ - 2}&5\\3&{ - 1}&6\end{array}} \right]\)

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).
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