Status: Tags: #cards/math232/unit4 Links: MATH 232 Units
Vector Coordinates
Principles
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- Coordinates ;; how much of each basis vector is in b
- Notation of basis of standard unit vectors ;; $B_o$
- (w)$B_o$ = ;; w
B as a set vs B as a matrix
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Textbook Definition
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System of linear equations in matrix form ? Bc = w, where B is basis vectors and c is constants
Find coordinates of a vector wrt a basis ?
- Set each vector of basis as a column, RHS will be the components of vector
- RREF
Theorem
Change of coordinates
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Change of coordinates matrix
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Let B = basis and S = standard basis. Change of coordinates matrix Q for B → S and matrix P for S → B ?
- Q = B → S would just be B
- P = S → B would equal the RREF of B in LHS and S in RHS
How to find change of coordinates matrix q from b-coord to c-coord, and the change of coordinates matrix p from c-coords to b-coords. verify pq = i
Examples
Find coordinates of v=(0,3) wrt basis B = {v1,v2}, v1=(2,1) and v2=(-1,1) ?
- I think bottom right component should be 1, pyke moment
- Each vector is a column, RREF and find valule sof each c
Convert coordinate vector of p(x) = 4 + x wrt to basis B = {-1 + 2x, 1 + x} ?
- Write p as linear combination of basis vectors in B
- Setup linear combination, distrivute and regroup
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- Plug into matrix, RHS = p(x) ?
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References:
Created:: 2022-02-20 20:33