MATH136 Lecture Notes - Lecture 24: Vector Space, Linear Combination
14 May 2018
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Friday, June 23
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Lecture 24 : Change of coordinates matrix II. (Refers to 4.3)
Concepts:
1. Computing the matrix which transforms the coordinates of a vector with respect
to a basis to coordinates of the same vector with respect to another basis.
24.1 Theorem – Let B = {b1, b2, …, bm} and C = {c1, c2, …, cm} be two bases of an
abstract vector space V of dimension m. Let x ∈ V. Then there exists a unique m by m
matrices CPB and BPC such that
CPB[x]B = [x]C
BPC[x]C = [x]B
The matrices CPB and BPC are as follows
Furthermore, CPBBPC = I.
Proof outline : Let B = {b1, b2, …, bm} and C = {c1, c2, …, cm} be two bases of an
abstract vector space V of dimension m. Suppose x belongs to V. Recall that T(x) = [x]B is
linear mapping and so [ax + by]B = a[x]B + b[y]B. Let [x]B = (a1, a2 …, am).
So CPB = [ [b1]C [b2]C [b3]C ... [bm]C ] maps [x]B to [x]C.
Similarly, BPC = [ [c1]B [c2]B [c3]B ... [cm]B ] maps [x]C to [x]B.
Also see that
Then
as required.
24.1.1 Definition – The matrix CPB is referred to as the change of coordinates matrix
from the B coordinates to the C coordinates.
Summarizing:
Remark: Note that the matrix CPB makes sense even when we are referring to abstract
vectors spaces in general, since its columns are column vectors.
24.2 Example – Suppose we are given two bases B = {b1, b2} and C = {c1, c2} for an
abstract vector space V. Suppose that
Suppose x is some vector in V such that x = 3b1 + b2. That is, suppose
Find [x]C .
Solution :
Altenratively:
24.2.1 Example – Suppose we are given two bases B = {b1, b2} and C = {c1, c2} for an
abstract vector space V. Suppose that
Suppose x is some vector in V such that x = 3b1 + b2. That is, suppose
Find [x]B .
Solution:
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Document Summary
Friday, june 23 lecture 24 : change of coordinates matrix ii. (refers to 4. 3) Concepts: computing the matrix which transforms the coordinates of a vector with respect to a basis to coordinates of the same vector with respect to another basis. The matrices cpb and bpc are as follows. So cpb = [ [b1]c [b2]c [b3]c [bm]c ] maps [x]b to [x]c. Similarly, bpc = [ [c1]b [c2]b [c3]b [cm]b ] maps [x]c to [x]b. 24. 1. 1 definition the matrix cpb is referred to as the change of coordinates matrix from the b coordinates to the c coordinates. Remark: note that the matrix cpb makes sense even when we are referring to abstract vectors spaces in general, since its columns are column vectors. 24. 2 example suppose we are given two bases b = {b1, b2} and c = {c1, c2} for an abstract vector space v. suppose that.
