Question

An odd Question about basis.

Answer 1

Assuming that v1,v2,.....,vn is a standard basis of \[R ^{n}\].

And there is a equation which is obviously right \[v1 = 1*v1+0*v2+0*v3+......+0*vn\]

Question: Is the v1 on the left hand side of the equation different from the v1 on the right hand side of the equation ?
MY PONIT is that the left v1 is a specific vector which can be written as (1,0,0,0,0,....,0),but the right v1 is an abstract vector which can not be wirtten as the form above.
Thus the left v1 is different to the right v2.

Answer 2

I have always thought of vectors as ordered tuples, and a basis vector can be any one in the set, though some vectors might be more convenient than others when serving as a basis vector. In this view, v1= 1*v1 + 0*v2 + ... means v1=v1, and the vectors are equal component by component. This does not really explain what <1,0> "means", other than "1 step in the first direction" and "0 steps in the 2nd direction". Your "abstract vector" idea corresponds to "first direction", but I never try to make much sense of that idea. In other words, I stick with ordered tuples, and basis vectors are the same as all other vectors.

Answer 3

All of which means I don't really know what "direction" means other than in some vague "operational" sense i.e. in the real world, I know a direction when I see one, but I can't explain it (and forget about explaining the difference between direction in space compared to direction in time.)

Answer 4

phi,Be grateful to you again. Your viewpoint is great and inspires me. The reason why I think of this question is that I feel confused when I try to understanding the idea that every linear map can be written as a matrix and linear map act like matrix mutiplication.