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Independent vectors and spanning vectors in a subspace of

Consider a subspace V of ${\mathbf{鈩�}}^{\mathbf{n}}$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 leastm vectors to span V.

c. If m vectors inV are linearly independent, then they form a basis of V.

d. If m vectors inV span V, then they form a basis of V.

Removing the dependent vectors$w_i$ from the set of vectors $B=\left\{{蠀}_1,{蠀}_2,....,{蠀}_p,w_1,w_2,...,w_j\right\}$.

Since,$w_j$ span V:

When they are linearly independent then they form a basis ofV and when they are linearly dependent then $j>dimV=m$.

Also, there is maximum linearly independent vector inV will be m. Thus, we need to eliminate the redundant vectors of$w_j$ fromB

As linearly independent vectors inV 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${\mathrm{鈩�}}^n$ remains valid.

Hence, proved that, a basis ofV consists of all the$v_i$ and some of the$w_j$