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If $\mathbf{a}$ and $\mathbf{b}$ are two non-zero and non-collinear vectors, then $\mathbf{a}+\mathbf{b}$ and $\mathbf{a}-\mathbf{b}$ are
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Linearly independent vectors
Since $\mathbf{a}$ and $\mathbf{b}$ are non-collinear, so $\mathbf{a}+\mathbf{b}$ and $\mathbf{a}-\mathbf{b}$ will also be non-collinear. Hence, $\mathbf{a}+\mathbf{b}$ and $\mathbf{a}-\mathbf{b}$ are linearly independent vectors.
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