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Linear Algebra and its Applications
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Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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Short Answer

Use Theorem 7 in section 1.7 to explain why the columns of the matrix A are linearly independent.

\(A = \left( {\begin{aligned}{*{20}{c}}1&0&0&0\\2&5&0&0\\3&6&8&0\\4&7&9&{10}\end{aligned}} \right)\)

The columns of matrix \(A\) are linearly independent according to theorem 7.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Denote the columns of matrix A

The columns of matrix \(A\) are denoted from right to left by vectors \({{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _4}\).

Step 2: Check if the columns of matrix A are linearly independent

Theorem 7 states that an indexed set \(S = \left\{ {{{\mathop{\rm v}\nolimits} _1},...,{v_p}} \right\}\) of two or more vectors is linearly dependent if and only if at least one of the vectors in \(S\) is a linear combination of the others. If \(S\) is linearly dependent and \({{\mathop{\rm v}\nolimits} _1} \ne 0\), then some \({{\mathop{\rm v}\nolimits} _j}\) is a linear combination of the preceding vectors \({{\mathop{\rm v}\nolimits} _1},...,{v_{j - 1}}\).

Vector \({{\mathop{\rm v}\nolimits} _1}\) is non-zero; \({{\mathop{\rm v}\nolimits} _2}\) is not a multiple of \({{\mathop{\rm v}\nolimits} _1}\) (since the third entry of \({{\mathop{\rm v}\nolimits} _2}\)is non-zero), and \({{\mathop{\rm v}\nolimits} _3}\) is not a linear combination of \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\) (since the second entry of \({{\mathop{\rm v}\nolimits} _3}\) is non-zero). Furthermore, \({{\mathop{\rm v}\nolimits} _4}\) cannot be a linear combination of \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},\) and \({{\mathop{\rm v}\nolimits} _3}\), based on the first entry in the vector. Thus, the columns are linearly independent according to theorem 7.

Most popular questions for Math Textbooks

Consider a dynamical system with two components. The accompanying sketch shows the initial state vector x→0 and two eigen vectors υ1→  and  υ2→ of A (with eigen values λ1→ and λ2→ respectively). For the given values of λ1→ and λ2→ , draw a rough trajectory. Consider the future and the past of the system.

λ1→=0.9,λ2→=0.9

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16: There exists a 2x2 matrix such that

A[11]=[12] and A[22]=[21] .

Construct a \(3 \times 3\) matrix\(A\), with nonzero entries, and a vector \(b\) in \({\mathbb{R}^3}\) such that \(b\) is not in the set spanned by the columns of\(A\).

Suppose the system below is consistent for all possible values of \(f\) and \(g\). What can you say about the coefficients \(c\) and \(d\)? Justify your answer.

27. \(\begin{array}{l}{x_1} + 3{x_2} = f\\c{x_1} + d{x_2} = g\end{array}\)

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Give separate answers for each part.

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