Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 1: Q10Q (page 1)

Let \({{\bf{a}}_1}\) \({{\bf{a}}_2}\), and b be the vectors in \({\mathbb{R}^{\bf{2}}}\) shown in the figure, and let \(A = \left( {\begin{aligned}{*{20}{c}}{{{\bf{a}}_1}}&{{{\bf{a}}_2}}\end{aligned}} \right)\). Does the equation \(A{\bf{x}} = {\bf{b}}\) have a solution? If so, is the solution unique? Explain.

Short Answer

Expert verified

The system of equations \(A{\bf{x}} = {\bf{b}}\) has a unique solution.

Step by step solution

01

Construct the graph with a grid

Consider the figure shown below:

On the \({x_1}{x_2}\)-plane, the lines between \({{\bf{a}}_1}\) and the origin and between \({{\bf{a}}_2}\) and the origin form a grid. Each point can be defined by using the grid.

02

Determine the solution

In the above figure,move some steps in the direction of the vectors to reach towards vectors\({{\bf{a}}_1}\),\({{\bf{a}}_2}\), and bfrom the origin.

There is always a unique way to reach these vectors. It means,\(A{\bf{x}} = {\bf{b}}\)has a solution.

Thus, the system of equations \(A{\bf{x}} = {\bf{b}}\) has a unique solution.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Give a geometric description of span \(\left\{ {{v_1},{v_2}} \right\}\) for the vectors \({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}8\\2\\{ - 6}\end{array}} \right]\) and \({{\mathop{\rm v}\nolimits} _2} = \left[ {\begin{array}{*{20}{c}}{12}\\3\\{ - 9}\end{array}} \right]\).

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

14: rank.|111123136|=3

In Exercises 11 and 12, determine if \({\rm{b}}\) is a linear combination of \({{\mathop{\rm a}\nolimits} _1},{a_2}\) and \({a_3}\).

12.

Let \(A = \left[ {\begin{array}{*{20}{c}}1&0&{ - 4}\\0&3&{ - 2}\\{ - 2}&6&3\end{array}} \right]\) and \(b = \left[ {\begin{array}{*{20}{c}}4\\1\\{ - 4}\end{array}} \right]\). Denote the columns of \(A\) by \({{\mathop{\rm a}\nolimits} _1},{a_2},{a_3}\) and let \(W = {\mathop{\rm Span}\nolimits} \left\{ {{a_1},{a_2},{a_3}} \right\}\).

  1. Is \(b\) in \(\left\{ {{a_1},{a_2},{a_3}} \right\}\)? How many vectors are in \(\left\{ {{a_1},{a_2},{a_3}} \right\}\)?
  2. Is \(b\) in \(W\)? How many vectors are in W.
  3. Show that \({a_1}\) is in W.[Hint: Row operations are unnecessary.]

If Ais an \(n \times n\) matrix and the equation \(A{\bf{x}} = {\bf{b}}\) has more than one solution for some b, then the transformation \({\bf{x}}| \to A{\bf{x}}\) is not one-to-one. What else can you say about this transformation? Justify your answer.a
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free