Now given this, let's see if weĬan come up with some other interesting ways in which these Number of linearly independent vectors you need to have aīasis for the orthogonal complement of V. Number of linearly independent vectors you need to haveĪ basis for V. Orthogonal complement of V, which is also another Learned that the dimension of V, plus the dimension of the Have even been the last video if I remember properly, we When you think about it this way, it is not so surprising that you can get anywhere in Rn using the direction information from V and V's orthogonal complement. V's orthogonal complement (with respect to Rn) contains the direction information to get to all the rest of Rn. The basis of V contains direction information that lets you get anywhere in V. The magnitudes of the basis vectors are not so important in this respect, because we can always scale them. I like to think of it in terms of "direction information". If you try to get to, you have to use +. Suppose V ⊆ ℝ³ is the xy plane and V⊥ ⊆ ℝ³ is the z axis. You need all the basis information from V and V⊥ to span ℝⁿ. V and w are orthogonal, because they are from orthogonal subspaces. To get to a particular x you have to use a particular v and w. Note that it does not mean you can use any v and any w. It means this: given V ⊆ ℝⁿ and V⊥ ⊆ ℝⁿ, you can find some v ∈ V to combine with some w ∈ V⊥ to make any x ∈ ℝⁿ. So does this mean that it takes only two vectors (as long as they are orthogonal) to span Rn for any n?
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