Essential Vocabulary

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Linear combination of vectors

A vector \(\vec {v}\) is said to be a linear combination of vectors \(\vec {v}_1, \vec {v}_2,\ldots , \vec {v}_n\) if
\[\vec {v}=a_1\vec {v}_1+ a_2\vec {v}_2+\ldots + a_n\vec {v}_n\]
for some scalars \(a_1, a_2, \ldots ,a_n\).

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Linearly dependent vectors

Let \(\vec {v}_1, \vec {v}_2,\ldots ,\vec {v}_k\) be vectors of \(\RR ^n\). We say that the set \(\{\vec {v}_1, \vec {v}_2,\ldots ,\vec {v}_k\}\) is linearly independent if the only solution to
\[c_1\vec {v}_1+c_2\vec {v}_2+\ldots +c_p\vec {v}_k=\vec {0}\]
is the trivial solution \(c_1=c_2=\ldots =c_k=0\).

If, in addition to the trivial solution, a non-trivial solution (not all \(c_1, c_2,\ldots ,c_k\) are zero) exists, then we say that the set \(\{\vec {v}_1, \vec {v}_2,\ldots ,\vec {v}_k\}\) is linearly dependent.

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Linearly independent vectors

Let \(\vec {v}_1, \vec {v}_2,\ldots ,\vec {v}_k\) be vectors of \(\RR ^n\). We say that the set \(\{\vec {v}_1, \vec {v}_2,\ldots ,\vec {v}_k\}\) is linearly independent if the only solution to
\[c_1\vec {v}_1+c_2\vec {v}_2+\ldots +c_p\vec {v}_k=\vec {0}\]
is the trivial solution \(c_1=c_2=\ldots =c_k=0\).

If, in addition to the trivial solution, a non-trivial solution (not all \(c_1, c_2,\ldots ,c_k\) are zero) exists, then we say that the set \(\{\vec {v}_1, \vec {v}_2,\ldots ,\vec {v}_k\}\) is linearly dependent.

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Redundant vectors

Let \(\{\vec {v}_1,\vec {v}_2,\dots ,\vec {v}_k\}\) be a set of vectors in \(\RR ^n\). If we can remove one vector without changing the span of this set, then that vector is redundant. In other words, if
\[\mbox {span}\left (\vec {v}_1,\vec {v}_2,\dots ,\vec {v}_k\right )=\mbox {span}\left (\vec {v}_1,\vec {v}_2,\dots ,\vec {v}_{j-1},\vec {v}_{j+1},\dots ,\vec {v}_k\right )\]
we say that \(\vec {v}_j\) is a redundant element of \(\{\vec {v}_1,\vec {v}_2,\dots ,\vec {v}_k\}\), or simply redundant.

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Span of a set of vectors

Let \(\vec {v}_1, \vec {v}_2,\ldots ,\vec {v}_p\) be vectors in \(\RR ^n\). The set \(S\) of all linear combinations of \(\vec {v}_1, \vec {v}_2,\ldots ,\vec {v}_p\) is called the span of \(\vec {v}_1, \vec {v}_2,\ldots ,\vec {v}_p\). We write
\[S=\mbox {span}(\vec {v}_1, \vec {v}_2,\ldots ,\vec {v}_p)\]
and we say that vectors \(\vec {v}_1, \vec {v}_2,\ldots ,\vec {v}_p\) span \(S\). Any vector in \(S\) is said to be in the span of \(\vec {v}_1, \vec {v}_2,\ldots ,\vec {v}_p\). The set \(\{\vec {v}_1, \vec {v}_2,\ldots ,\vec {v}_p\}\) is called a spanning set for \(S\).

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