- #1
Dell
- 590
- 0
givem,-W is the group of all the vectors in R4 which are orthagonal to the vectors v1(1 0 1 0) and also v2(1 2 1 1)
a) find a basis and the dimention of W
b) find, in W, a vector which is orthagonal to the vector (1 -2 1 3)
a)
i call this vector w=(a b c d) and say
v1(dot)w=0
v2(dot)w=0
therefore
a+0b+c+0d=0 a=-c
a+2b+c+d=0 d=-2b
i will have 2 free parameters here, 4 unknows -rank2=2
a=t
d=u
so w=(t, -2u, -t, u)
W=sp{(1 0 -1 0), (0 -2 0 1)}
dim(w)=2
----------------------------
b)
(t, -2u, -t, u) dot (1 -2 1 3)=0
t+4u-t+3u=0
7u=0
u=0
therefore any vector that fits
(t 0 -t 0) is orthagonal to (1 -2 1 3) and a part of W
is this all correct
a) find a basis and the dimention of W
b) find, in W, a vector which is orthagonal to the vector (1 -2 1 3)
a)
i call this vector w=(a b c d) and say
v1(dot)w=0
v2(dot)w=0
therefore
a+0b+c+0d=0 a=-c
a+2b+c+d=0 d=-2b
i will have 2 free parameters here, 4 unknows -rank2=2
a=t
d=u
so w=(t, -2u, -t, u)
W=sp{(1 0 -1 0), (0 -2 0 1)}
dim(w)=2
----------------------------
b)
(t, -2u, -t, u) dot (1 -2 1 3)=0
t+4u-t+3u=0
7u=0
u=0
therefore any vector that fits
(t 0 -t 0) is orthagonal to (1 -2 1 3) and a part of W
is this all correct