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Expert-verified Found in: Page 132 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # Question: Consider linearly independent vectors $\stackrel{\mathbf{\to }}{{\mathbf{v}}_{\mathbf{1}}}{\mathbf{,}}\stackrel{\mathbf{\to }}{{\mathbf{v}}_{\mathbf{2}}}{\mathbf{,}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{,}}\stackrel{\mathbf{\to }}{{\mathbf{v}}_{\mathbf{m}}}$ in ${{\mathbf{ℝ}}}^{{\mathbf{n}}}$ and let A be an invertible ${\mathbit{m}}{\mathbf{×}}{\mathbit{m}}$matrix. Are the columns of the following matrix linearly independent?$\left[\begin{array}{ccc}\frac{I}{{V}_{1}}& \frac{I}{{V}_{2}}...& \frac{I}{{V}_{m}}\\ I& I& I\end{array}\right]{\mathbit{A}}$

Yes, the columns of the matrix linearly independent.

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

See the step by step solution

## Step 1: Definition of Linearly independent

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.

## Step 2: Prove the vectors are linearly independent

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

We are given a $m×m$ 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.

## Step 3: The final answer

Yes, the columns of the matrix are linearly independent.

$\left[\begin{array}{ccc}\frac{\mathrm{I}}{{\mathrm{V}}_{1}}& \frac{\mathrm{I}}{{\mathrm{V}}_{2}}...& \frac{\mathrm{I}}{{\mathrm{V}}_{m}}\\ \mathrm{I}& \mathrm{I}& \mathrm{I}\end{array}\right]A$ ### Want to see more solutions like these? 