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Q67E

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Found in: Page 145

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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

Hence, a basis of V consists of all the ${v}_{i}$ and some of the ${w}_{j}$.

See the step by step solution

## Step 1: Define the basis.

Independent vectors and spanning vectors in a subspace of

Consider a subspace V of ${{\mathbf{ℝ}}}^{{\mathbf{n}}}$ with ${\mathbit{d}}{\mathbit{i}}{\mathbit{m}}\left(V\right){\mathbf{=}}{\mathbit{m}}$.

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 $B=\left\{{\upsilon }_{1},{\upsilon }_{2},....,{\upsilon }_{p},{w}_{1},{w}_{2},...,{w}_{j}\right\}$.

Since,${w}_{j}$ span V:

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

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 $j-\left(m-p\right)$ vectors.

Therefore, we have to eliminate some vectors ${w}_{j}$ so that the condition of independent vectors and spanning vectors in a subspace of ${\mathrm{ℝ}}^{n}$ 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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