Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Consider linearly independent vectors ${\vec{υ}}_{1}, {\vec{υ}}_{2}, . . ., {\vec{υ}}_{p}$ in a subspace V of $ℝ^{n}$ and vectors $w_{1}, w_{2}, . . ., w_{q}$ that span V. Show that there is a basis of V that consists of all the ${\vec{u}}_{i}$ and some of the $\vec{w_{j}}$ . Hint: Find a basis of the image of the matrix
$A = [\begin{matrix} | & | & | & | \\ {\vec{v}}_{1} & . . . & {\vec{v}}_{1} & {\vec{w}}_{1} & . . . & {\vec{w}}_{q} \\ | & | & | & | \end{matrix}]$

Hence, a basis of V consists of all the $v_{i}$ and some of the $w_{j}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Define the basis.

Independent vectors and spanning vectors in a subspace of

Consider a subspace V of $ℝ^{n}$ with $d i m (V) = m$ .

a. We can find at most m linearly independent vectors in V.

b. We need at least m vectors to span V.

c. If m vectors in V are linearly independent, then they form a basis of V.

d. If m vectors in V span V, then they form a basis of V.

Step 2: Prove that a basis of V consists of all the v→i and some of the w→j.

Removing the dependent vectors $w_{i}$ from the set of vectors $B = \{υ_{1}, υ_{2}, . . . ., υ_{p}, w_{1}, w_{2}, . . ., w_{j}\}$ .

Since, $w_{j}$ span V:

When they are linearly independent then they form a basis of V and when they are linearly dependent then $j > d i m V = m$ .

Also, there is maximum linearly independent vector in V will be m. Thus, we need to eliminate the redundant vectors of $w_{j}$ from B

As linearly independent vectors in V form a basis. So, we have to eliminate $j - (m - p)$ vectors.

Therefore, we have to eliminate some vectors $w_{j}$ so that the condition of independent vectors and spanning vectors in a subspace of $ℝ^{n}$ remains valid.

Hence, proved that, a basis of V consists of all the $v_{i}$ and some of the $w_{j}$

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Linear Algebra With Applications

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Short Answer

Step by Step Solution

Step 1: Define the basis.

Step 2: Prove that a basis of V consists of all the v→i and some of the w→j.

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Linear Algebra With Applications

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Short Answer

Consider linearly independent vectors υ→1,υ→2,...,υ→p in a subspace V of ℝn and vectors w1,w2,...,wq that span V. Show that there is a basis of V that consists of all the u→i and some of the wj→. Hint: Find a basis of the image of the matrixA=[| || |v→1...v→1w→1...w→q| || |]

Step by Step Solution

Step 1: Define the basis.

Step 2: Prove that a basis of V consists of all the v→i and some of the w→j.

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