Chapter 3, Problem 1E

If ${\stackrel{\to }{v}}_1,{\stackrel{\to }{v}}_2,\mathrm{...},{\stackrel{\to }{v}}_n$ and ${\stackrel{\to }{w}}_1,{\stackrel{\to }{w}}_2,\mathrm{...},{\stackrel{\to }{w}}_m$ are any twobases of a subspace Vof ${ℝ}^{10}$ , then n must equal m.

To determine

To find: Whether the given statement is true or false.

Answer to Problem 1E

The given statement is true.

Explanation of Solution

Given information: The statement is “If $\overrightarrow{v_1},\overrightarrow{v_2},\mathrm{....}\overrightarrow{v_n}$ and $\overrightarrow{w_1},\overrightarrow{w_2},\mathrm{....}\overrightarrow{w_n}$ are any two bases of a subspace $V$ of $R^{10}$ , then $n$ must equal $m$ .

The dimension of subspace is equal to the number of vectors in any basis for the subspace. Simply, all the basis of subspace has equal number of vectors.

Consider, the subspace $V$ of $R^{10}$ having two bases, $\overrightarrow{v_1},\overrightarrow{v_2},\mathrm{....}\overrightarrow{v_n}$ and $\overrightarrow{w_1},\overrightarrow{w_2},\mathrm{....}\overrightarrow{w_n}$ .

So, the two bases are having same number of vectors. That is $n=m$

Hence, the statement is true.