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### Linear Algebra With Applications

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

# Consider a subspace ${\mathbit{V}}$ in ${{\mathbit{ℝ}}}^{{\mathbf{m}}}$ that is defined by homogeneous linear equations$|\begin{array}{l}{a}_{11}{x}_{1}+{a}_{12}{x}_{2}+\dots +{a}_{1m}{x}_{m}=0\\ {a}_{21}{x}_{1}+{a}_{22}{x}_{2}+\dots +{a}_{2m}{x}_{m}=0\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}⋮\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}⋮\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}⋮\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}⋮\text{\hspace{0.17em}}\\ {a}_{n1}{x}_{1}+{a}_{22}{x}_{2}+\dots +{a}_{\mathrm{nm}}{x}_{m}=0\end{array}|$.What is the relationship between the dimension of ${\mathbit{V}}$and the quantity ${\mathbit{m}}{\mathbf{-}}{\mathbit{n}}$? State your answer as an inequality. Explain carefully.

A subspace $V$ of ${ℝ}^{m}$ has dimension at least $m-n$.

See the step by step solution

## Step 1: To mention given data and recall the theorem 3.3.7

We have the subspace $V$ of ${ℝ}^{m}$ defined by $n$ homogeneous linear eqautions,

role="math" localid="1660126682574" $\left|\begin{array}{l}{a}_{11}{x}_{1}+{a}_{12}{x}_{2}+\cdots +{a}_{1m}{x}_{m}=0\\ {a}_{21}{x}_{1}+{a}_{22}{x}_{2}+\cdots +{a}_{2m}{x}_{m}=0\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}⋮\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}⋮\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}⋮\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}⋮\\ {a}_{n1}{x}_{1}+{a}_{n2}{x}_{2}+\cdots +{a}_{nm}{x}_{m}=0\end{array}\right|$,

Therefore, $V$ can be written as

$V=\mathrm{ker}\left(A\right)$, where $A$ is an $n×m$ matrix $A=\left[{a}_{ij}\right]$ .

Suppose the column vector be $\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\\ ⋮\\ {x}_{m}\end{array}\right]$ .

Theorem 3.3.7, is stated as follows:

For any $n×m$ matrix $A$, the equation,

$\mathrm{dim}\left(\mathrm{ker}\left(\mathrm{A}\right)\right)+\mathrm{dim}\left(\mathrm{im}\left(\mathrm{A}\right)\right)=\mathrm{m}$ .

## Step 2: To find the relationship between the dimension V and m-n

We have the rank of $A$ $rank\left(A\right)\le n$.

Thus, by Theorem 3.3.7, we have,

$\begin{array}{c}\mathrm{dim}\left(\mathrm{V}\right)=\mathrm{dim}\left(\mathrm{ker}\left(\mathrm{A}\right)\right)\\ =\mathrm{m}-\mathrm{rank}\left(\mathrm{A}\right)\\ \ge \mathrm{m}-\mathrm{n}\end{array}$

$\therefore \mathrm{dim}\left(\mathrm{V}\right)\ge \mathrm{m}-\mathrm{n}$

.

Hence, a subspace $V$ of ${\mathrm{ℝ}}^{\mathrm{m}}$ has dimension at least $m-n$.