# Thread: Solve the matrix problem

1. ## Solve the matrix problem

I have no idea of this question, plz help me !~thanks

Let H be the span of the following six vextors:
V1=[1,-1,0,0]
V2=[1,0,-1,0]
V3=[1,0,0,-1]
v4=[0,1,-1,0]
v5=[0,1,0,-1]
v6=[0,0,1,-1]

1.Show that B={V1,V2,V3} is a basis for H .Do this by showing that (a) the vextors in B are independent, and (b) the remaining three vectors canbe each be expressed as a linear combination of vectors in B.

2. Find a different subset of the vectors that is still a basis for H.

3.Find a non-zero 1*4 matrix A such that Av=0 for all V in H

2. Anyone help me?!!!~
Originally Posted by shannon1111
I have no idea of this question, plz help me !~thanks

Let H be the span of the following six vextors:
V1=[1,-1,0,0]
V2=[1,0,-1,0]
V3=[1,0,0,-1]
v4=[0,1,-1,0]
v5=[0,1,0,-1]
v6=[0,0,1,-1]

1.Show that B={V1,V2,V3} is a basis for H .Do this by showing that (a) the vextors in B are independent, and (b) the remaining three vectors canbe each be expressed as a linear combination of vectors in B.

2. Find a different subset of the vectors that is still a basis for H.

3.Find a non-zero 1*4 matrix A such that Av=0 for all V in H

3. Originally Posted by shannon1111
I have no idea of this question, plz help me !~thanks

Let H be the span of the following six vextors:
V1=[1,-1,0,0]
V2=[1,0,-1,0]
V3=[1,0,0,-1]
v4=[0,1,-1,0]
v5=[0,1,0,-1]
v6=[0,0,1,-1]

1.Show that B={V1,V2,V3} is a basis for H .Do this by showing that (a) the vextors in B are independent, and (b) the remaining three vectors canbe each be expressed as a linear combination of vectors in B.

2. Find a different subset of the vectors that is still a basis for H.

3.Find a non-zero 1*4 matrix A such that Av=0 for all V in H
you start panicking if you don't get a response in 1 and a half hours?

Did you spend any of that time working on the problem?

For 1(a) look at a a[1,-1,0,0]+ b[1,0,-1,0]+c[1,0,0,-1]= [0, 0, 0, 0] and show that a, b, c must all be 0. That's the definition of "independent" isn't it?

For 1(b) look at a[1,-1,0,0]+ b[1,0,-1,0]+c[1,0,0,-1]= [0,1,-1,0] and show that you can find a, b, and c so that is true. Do the same for the other vectors.

For 2, just pick out three vectors and see if they are independent (V4, V5, V6 clearly won't work because of that "0" in the first place of each so you need at least one of V1, V2, V3 with other vectors).

For 3, look at [a, b, c, d][1, ,-1, 0, 0]= a- b= 0, [a, b, c, d][1, 0, -1, 0]= a- c= 0, and [a, b, c, d][1, 0, 0, -1]= a- d= 0. If a linear transformation makes every vector in a basis 0, it makes every vector in their span 0.