Question

An Question about Linear Algebra

Assuming that v1,v2,.....,vn is a standard basis of Rn

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 1

They are the same.

Answer 2

any vector in \(\mathbb R^n\) can be expressed as combination of the basis, right?

Answer 3

so, why not \(\vec v_1\) itself?

Answer 4

I take it down to \(\mathbb R^2\)
Let say, \(\{\left[\begin{matrix}1\\0\end{matrix}\right]\left[\begin{matrix}0\\1\end{matrix}\right]\},\)is the standard base of \(\mathbb R^2\)

Answer 5

Then, any vector in R2 can be expressed as combination of them. for example
\(\left[\begin{matrix}2\\3\end{matrix}\right]=2\left[\begin{matrix}1\\0\end{matrix}\right]+3\left[\begin{matrix}0\\1\end{matrix}\right]\)
right?

Answer 6

hence, if my vector is \(\left[\begin{matrix}1\\0\end{matrix}\right]\), then it is
\(\left[\begin{matrix}1\\0\end{matrix}\right]=1\left[\begin{matrix}1\\0\end{matrix}\right]+0\left[\begin{matrix}0\\1\end{matrix}\right]\)

Answer 7

then, the left vector and the right vector are the same.
expand the logic to \(\mathbb R^n\)

Answer 8

but how \[\left(\begin{matrix}1 \\0\end{matrix}\right) and \left(\begin{matrix}0 \\1\end{matrix}\right)\] be defined? I am confused.

Answer 9

(1,0) and (0,1) are the standard basis of R2.

Answer 10

Whenever you talk about "STANDARD", you indicate to 1 at \(a_{ij}\) where i =j and 0 if \(i\neq j\)

Answer 11

in R3, the standard basis is