show that the standard unit vector e1, e2,e3......en are linearly independent in Rn, sorry for 1,2,3 and n there are supposed to be subsripts

Question

show that the standard unit vector e₁, e₂,e₃......e_n are linearly independent in Rⁿ

Answer 1

As standard unit vectors, 

 

e₁ = (1,0,...,0)

e₂ = (0,1,...,0)

...

e_n = (0,0,...1)

 

There are a couple of ways to approach this depending on how vigorous of a proof class this is.  One thing to note is that (e₁, e₂, ...,e_n) form a basis for Rⁿ.  Therefore they are linearly independent. 

 

In general, vectors e1, e2, ..., e_n are linearly independent if:

a₁*e₁ + a₂*e₂ + ... + a_n*e_n = (0,0,...,0) implies that a₁ = a₂ = ... = a_n = 0

 

Since only e1 has a non-zero number as the first entry, a₁ = 0.

Since only e2 has a non-zero number as the second entry, a₂ = 0. 

...

Since only en has a non-zero number as the n^th entry, a_n = 0.

 

Therefore e₁, e₂, ..., e_n are linearly independent.

 

 

J.T.