Some concepts on linear algebra that have to make clear

[SlowLearner007]

Dear everyone, l really get confused when reading some textbooks on linear algebra :

Suppose there are two vectors in R3, u= (3,1,0) and v = (1,6,0).
Firstly, u and v are linearly independent because neither vector is a multiple of the other.
IF w is a linear combination of u and v, then {u,v,w} is linearly dependent and w is in span {u,v}.

So , it joined to conclusion that any set {u,v,w} in R3 with u, and v linearly independent. The set
{u,v,w} will be linearly dependent if and only if w is in the plane spanned by u and v .

l wonder if this theorem is true if x₃ is not equal to 0 for the above case.

Besides, are the following statements true ?
1) If x and y are linearly independent, and if {x,y,z} is linearly dependent, then z is in Span {x,y}
2) lf x and y are linearly independent , and if z is in Span {x,y}, then {x,y,z} is linearly dependent.

Now for a practice problem:
There are four vectors u= (3,2,-4)
v= (-6,1,7)
w= (0,-5,2)
z= (3,7, -5)
It shows that each pairs of the above vectors are linearly independent because neither vector is a multiple of the other.
But, {u,v,w,z} is linearly dependent, because there are more vectors than entries in them.

So my question arised, how coould{u,v,w,z} be linearly dependent when w is NOT in span{u,v,z} ?

 

[HallsofIvy]

Dear everyone, l really get confused when reading some textbooks on linear algebra :

Suppose there are two vectors in R3, u= (3,1,0) and v = (1,6,0).
Firstly, u and v are linearly independent because neither vector is a multiple of the other.
IF w is a linear combination of u and v, then {u,v,w} is linearly dependent and w is in span {u,v}.

So , it joined to conclusion that any set {u,v,w} in R3 with u, and v linearly independent. The set
{u,v,w} will be linearly dependent if and only if w is in the plane spanned by u and v .

l wonder if this theorem is true if x₃ is not equal to 0 for the above case.

Besides, are the following statements true ?
1) If x and y are linearly independent, and if {x,y,z} is linearly dependent, then z is in Span {x,y}
2) lf x and y are linearly independent , and if z is in Span {x,y}, then {x,y,z} is linearly dependent.

Now for a practice problem:
There are four vectors u= (3,2,-4)
v= (-6,1,7)
w= (0,-5,2)
z= (3,7, -5)
It shows that each pairs of the above vectors are linearly independent because neither vector is a multiple of the other.
But, {u,v,w,z} is linearly dependent, because there are more vectors than entries in them.

So my question arised, how coould{u,v,w,z} be linearly dependent when w is NOT in span{u,v,z} ?

Well, that can happen if u, v, and z are themselves NOT linearly independent. Did you check that?

However, here I get that w is in the span of u, v, and z (w= (11/9)u+ (1/9)v- z) so I do not accept your assertion!.