### Projection onto Linearly Dependent Vectors

Now consider another example:

The projections of onto these vectors are

The sum of the projections is

Something went wrong, but what? It turns out that a set of vectors can be used to reconstruct an arbitrary vector in from its projections only if they are linearly independent. In general, a set of vectors is linearly independent if none of them can be expressed as a linear combination of the others in the set. What this means intuitively is that they must point in different directions'' in -space. In this example so that they lie along the same line in -space. As a result, they are linearly dependent: one is a linear combination of the other ( ).

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