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Q67E

Expert-verifiedFound in: Page 145

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**Consider linearly independent vectors ${\overrightarrow{\mathbf{\upsilon}}}_{{\mathbf{1}}}{\mathbf{,}}{\overrightarrow{\mathbf{\upsilon}}}_{{\mathbf{2}}}{\mathbf{,}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{,}}{\overrightarrow{\mathbf{\upsilon}}}_{{\mathbf{p}}}$ in a subspace V of ${{\mathbf{\mathbb{R}}}}^{{\mathbf{n}}}$ 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 ${\overrightarrow{\mathbf{u}}}_{{\mathbf{i}}}$ and some of the $\overrightarrow{{\mathbf{w}}_{\mathbf{j}}}$. Hint: Find a basis of the image of the matrix**

${\mathit{A}}{\mathbf{=}}{\left[\begin{array}{cccccc}|& & |& |& & |\\ {\overrightarrow{v}}_{1}& ...& {\overrightarrow{v}}_{1}& {\overrightarrow{w}}_{1}& ...& {\overrightarrow{w}}_{q}\\ |& & |& |& & |\end{array}\right]}$

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

**Independent vectors and spanning vectors in a subspace of **

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

**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.**

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 $j-(m-p)$ vectors.

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