12. Find a basis for all of the vectors in lR^2 that lie on the line style=\"fon
ID: 2986953 • Letter: 1
Question
12. Find a basis for all of the vectors in lR^2 that lie on the linestyle="font-size: 12pt; font-family: arialmt; ">style="font-size: 12pt; font-family: tci4; ">style="font-size: 12pt; font-family: arialmt; ">
y = 4xstyle="font-size: 12pt; font-family: timesnewromanpsmt; ">style="font-size: 12pt; font-family: tci1; ">style="font-size: 12pt; font-family: timesnewromanps; font-style: italic; ">
13. What is the "standard basis" for P2 and how would you write the following vectorstyle="font-size: 12pt; font-family: arialmt; ">style="font-size: 12pt; font-family: timesnewromanps; font-style: italic; ">style="font-size: 12pt; font-family: arialmt; ">style="font-size: 12pt; font-family: timesnewromanpsmt; ">
from P2 in terms of that basisstyle="font-size: 12pt; font-family: arialmt; ">style="font-size: 12pt; font-family: timesnewromanps; font-style: italic; ">style="font-size: 12pt; font-family: arialmt; ">
p(t) 3 - 4 t+ 5t2style="font-size: 12pt; font-family: timesnewromanpsmt; ">style="font-size: 12pt; font-family: timesnewromanps; font-style: italic; ">
14. Is the following collection of vectors also a basis for P2 and why? If it is a basisstyle="font-size: 12pt; font-family: arialmt; ">style="font-size: 12pt; font-family: timesnewromanps; font-style: italic; ">style="font-size: 12pt; font-family: arialmt; ">style="font-size: 12pt; font-family: timesnewromanpsmt; ">
what is the change of coordinate matrix from this basis to the standard basis.style="font-size: 12pt; font-family: arialmt; ">
B ={ 1- 3t^2, 2 + t - 5t^2, 1+2t }style="font-size: 12pt; font-family: tci3; ">face="tci1">style="font-size: 12pt; font-style: italic; ">style="font-size: 12pt; font-family: timesnewromanps; font-style: italic; ">
Explanation / Answer
12)
the line y = 4x has the basis = { 4 }
13) Standard basis for P2 = { 1, t , t^2 }
p(t) = 3*(1) -4 *(t) + 5 * (t^2)
14) To check for the basis , find the linear dependency of the vectors.
det [ (-3,0,1),( -5,1,2), (0,2,1) ] = -19 != 0
Hence they are linear independent and hence form a basis.
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