2 Let H span 1 1 0 1 0 2 3 4 2 0 3 2 2 2 0 2 Determine a ba

Let H = span {[1 1 0 -1], [0 -2 3 4], [2 0 3 2], [-2 -2 0 2]}. Determine a basis for H. Give the dimension of H. Let T: R^n rightarrow R^m be a linear, 1-1 map. Prove: If {v_1, v_2,v_3} is a linearly independent set in R^m.

Solution

2. Denote the vectors given in order as : a,b,c,d

So,

d=-2a

c=b+2a

So, b and a span H

So basis for H is {a,b}

Dimension is number of vectors in basis

Hence dimension of H is 2

3.

Let, a,b,c so that

aT(v1)+bT(v2)+cT(v3)=0

Since T is linear it become

T(av1+bv2+cv3)=0

But T is 1-1 map hence

av1+bv2+cv3=0

But v1,v2,v3 is a linearly indepdent set

Hence, a=b=c=0

Hence, T(v1),T(v2),T(v3) is a linearly independent set.