Coordinate systems
The main reason for selecting a basis for a subspace H can be written in only one way as a linear combination of the basis vectors. To see why, suppose B = {b₁,....,bp} is a basis for H, and suppose a vector x in H can be generated in two ways say,
x=c₁b₁+·····+cpbp and x=d₁b₁+····+dpbp (1)
Then subtracting gives
0=x-x =(c₁-d₁)b₁+·······+(cp-dp) bp (2)
Since B is linearly independent, the weights in (2) must all be zero. That is, cⱼ=dⱼ for 1≤j≤p, which shows that the two representations in (1) are actually the same.
The invertible Matrix theorems
Let A be an nxn matrix. then the following statements are each equivalent to the statement that A is an invertible matrix.
1. The columns of form a basis of Rⁿ.
2. Col A= Rⁿ
3. dim Col A= n
4. rank A= n
5. Nul A= {0}
6. dim Nul A= 0
The rank theorem
If a matrix A has n columns, then rank A+ dim Nul A= n.
The basis theorem
Let H be a p-dimensional subspace of Rⁿ. Any linearly independent set of exactly p elements in H is automaticaly a basis for H. Also, any set of p elements of H that spans H is automaticaly a basis for H.
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