Let M1 1 1 1 0 M2 1 1 1 0M3 1 1 1 0 M4 1 1 1 0 Let U sp

Let M_1 = [1 1 1 0], M_2 = [1 1 -1 0],M_3 = [1 -1 1 0], M_4 = [-1 -1 1 0] Let U = span{M_1, M_2, M_3}· What is dim(U)? Let W = span {M_1, M_2,M_3, M_4}. What is dim(W)?

Solution

a) Consider M1, M2, M3

To find dim (U) we check whether the 3 are linearly independent,.

If possible let M1 = aM2+bM3

Then equate corresponding elements

1 =a+b

1=-a+b

Solving, b =1 and a =0

Also 1 =a-b = 0 which is a contradiction

So M1, M2 and M3 are linearly independent.

Dim U = dim span (M1,M2,M3) = 3 as 3 vectors are linearly independent.

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b)W= span of M1, M2, M3, M4

We already proved M1, M2 and M3 are linearly independent

]Now checking for M4

Let M4 = aM1+bM2+cM3

Equate corresponding elements

-1=a+b+c: 1=a-b+c; -1 = a+b-c

0=0

There are 3 equations and 3 variables

Solving we have M4 = -M2 since a =c=0 and b =-1

Hence M1, M2,M3 and M4 are not linearly independent.

So dim W = dim span M1, M2, M3 = 3