MATH136 Lecture Notes - Lecture 23: Linear Combination
14 May 2018
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Wednesday, June 21 − Lecture 23 : Change of coordinates matrix I. (Refers to 4.3)
Concepts:
1. Recall: When expressing a vector v in V as a linear combination of the vectors in
a basis, the coefficients are unique.
2. Define and find the coordinates of v with respect to a given basis B of a vector
space V.
23.1 Recall − The unique representation theorem. If V is a vector space and
B = { v1, v2, .... , vk } forms a basis for V then, for every v in V there is a unique set
of coefficients α1, α2, ..., αk in ℝ such that v = α1v1+ α2v2 + .... + αkvk.
23.2 Definition − If v is a vector in the vector space V which has an ordered basis
B = {v1, v2, ...., vk }. Then the unique coefficients α1, α2, ..., αk used to express v
as a linear combination of B = {v1, v2, ...., vk} are called the coordinates of v with
respect to the basis B.
We will represent these k unique scalars α1, α2, ...., αk by [v]B , i.e.,
[v]B = (α1, α2, ...., αk).
Since [v]B is expressed as a k-tuple, the order of αi’s must respect the order in which the
elements of the basis B ={v1, v2, ...., vk } are given. We refer to B as an ordered basis.
23.2.1 Examples
a) Suppose S = {e1, e2} is the standard basis of V = ℝ2. Consider v = (−1, 7).
Since (−1, 7) = −1e1 + 7e2, then the unique coordinates [v]S of v with respect to
the basis S are
[v]S = [(−1,7)] S = (−1, 7).
The coordinates of v with respect to the standard basis S of are ℝn are called the
standard coordinates. Notice that [v]S = v. Equality [v]S = v holds true only when
the vector space V = ℝn.
b) Suppose we have another basis B = {u1, u2} for ℝ2, where u1 = (1, 2) and
u2 = (3, 5).
Find the unique coordinates [v]B of v = (−1,7) with respect to the basis B.
Solution: We solve for
since [v]B = (α, β). Solving the matrix equation is equivalent to solving for
or
to obtain as unique solution α = 26 and β = −9.
Notation – If we denote the matrix [u1, u2] as PB , then PB maps [v]B to v.
23.3 Definition − Suppose B = { b1, b2, ...., bm } is an ordered basis of ℝm. The matrix
whose columns are the vectors of the basis B = { b1, b2, ...., bm } is called coordinate
matrix with respect to the ordered basis B. We will denote it by
PB = [b1 b2 .... bm ]
The matrix PB is seen as mapping [v]B to v.
Computing [v]B is obtained by solving the system
[ PB | v ] = [ b1 b2 .... bm | v ].
23.3.1 Example − Consider the basis B = {(1, 0, 0), (3, 1, 0), (1, 1, 1)} = { b1, b2, b3 } of ℝ3.
Find the coordinates of u1 = (3, 4, 5) and of u2 = (2, 9, 5) with respect to
the basis B.
Solution:
We first construct the coordinates matrix SPB. The matrix PB is the matrix whose
columns are the vectors in the ordered basis B (respecting the order). We have
We must solve the two systems:
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Document Summary
Wednesday, june 21 lecture 23 : change of coordinates matrix i. (refers to 4. 3) 23. 2 definition if v is a vector in the vector space v which has an ordered basis. Then the unique coefficients 1, 2, , k used to express v as a linear combination of b = {v1, v2, , vk} are called the coordinates of v with respect to the basis b. We will represent these k unique scalars 1, 2, , k by [v]b , i. e. , Since [v]b is expressed as a k-tuple, the order of i"s must respect the order in which the elements of the basis b ={v1, v2, , vk } are given. We refer to b as an ordered basis. Since ( 1, 7) = 1e1 + 7e2, then the unique coordinates [v]s of v with respect to the basis s are: suppose s = {e1, e2} is the standard basis of v = 2.
