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Q44E

Expert-verifiedFound in: Page 132

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**Question: Consider linearly independent vectors $\overrightarrow{{\mathbf{v}}_{\mathbf{1}}}{\mathbf{,}}\overrightarrow{{\mathbf{v}}_{\mathbf{2}}}{\mathbf{,}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{,}}\overrightarrow{{\mathbf{v}}_{\mathbf{m}}}$**** in ${{\mathbf{\mathbb{R}}}}^{{\mathbf{n}}}$ and let A be an invertible ${\mathit{m}}{\mathbf{\times}}{\mathit{m}}$****matrix. Are the columns of the following matrix linearly independent?**

${\left[\begin{array}{ccc}\frac{I}{{V}_{1}}& \frac{I}{{V}_{2}}...& \frac{I}{{V}_{m}}\\ I& I& I\end{array}\right]}{\mathit{A}}$

Yes, the columns of the matrix linearly independent.

$\left[\begin{array}{ccc}\frac{\mathrm{I}}{{\mathrm{V}}_{1}}& \frac{\mathrm{I}}{{\mathrm{V}}_{2}}...& \frac{\mathrm{I}}{{\mathrm{V}}_{\mathrm{m}}}\\ \mathrm{I}& \mathrm{I}& \mathrm{I}\end{array}\right]A$

**Linearly independent is defined as the property of a set having no linear combination of its elements equal to zero when the coefficients are taken from a given set unless the coefficient of each element is zero.**

$\overrightarrow{{\mathrm{v}}_{1}},\overrightarrow{{\mathrm{v}}_{2}},....,\overrightarrow{{\mathrm{v}}_{\mathrm{m}}}$ are linearly independent vectors.

We are given a $m\times m$ matrix A that is invertible.

Let’s say that

$\mathrm{B}=\left[\begin{array}{ccc}\frac{\mathrm{I}}{{\mathrm{V}}_{1}}& \frac{\mathrm{I}}{{\mathrm{V}}_{2}}...& \frac{\mathrm{I}}{{\mathrm{V}}_{\mathrm{m}}}\\ \mathrm{I}& \mathrm{I}& \mathrm{I}\end{array}\right]$

Since they are independent we know that ker B = 0

Therefore kerBA = 0 so columns of AB are linearly independent.

Yes, the columns of the matrix are linearly independent.

$\left[\begin{array}{ccc}\frac{\mathrm{I}}{{\mathrm{V}}_{1}}& \frac{\mathrm{I}}{{\mathrm{V}}_{2}}...& \frac{\mathrm{I}}{{\mathrm{V}}_{m}}\\ \mathrm{I}& \mathrm{I}& \mathrm{I}\end{array}\right]A$

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