linear algebra
1) Does the following set of polynomials form a basis of ??4? (Justify) ( 3 points)
{ 2, 2 – 3??
2
, 2 – 3??
2 – 5??
3
, 2 – 4??
2 – 5??
4
, 2 – 4??
3 – 5??
4
}
2) Let A be a 4??7 matrix. Does the equation ??
???? = ?? has solution for every ?? ???????????? ??
7
? (Justify)
( 2 points)
3) Let B = {??1, ??2, ??3
} , ?? = {??1, ??2, ??3
} are two basis of ??
3
. Find the change of co-ordinate matrix
from B to C, and then find the change of co-ordinate matrix from C to B. ( 5 points)
??1 = [
1
0
0
], ??2 = [
1
1
0
], ??3 = [
1
1
1
], ??1 = [
1
2
3
], ??2 = [
2
2
3
], ??3 = [
0
0
1
]
4) In ??3 find the change of co-ordinate matrix from the basis
B = { 1, 1 + ??, 1 + ?? + ??
2
, 1 + ?? + ??
2 + ??
3
} to the standard basis. Then write 2 + 4?? + ??
3
as a
linear combination of the polynomials in B. ( 5 points)
