linear algebra (vector and spaces), column space.

[NotBright]

.I have been trying to understand this question and solve it for several days. Yet i'm not getting any wiser. The question is getting harder and harder each time i'm trying to understand it. I'm a high school student.

Question; Which of the following vectors Belong to the column space of matrix A
A(3x3):
(0 1 2)
(1 1 1)
(-1 2 5)

a) (0,1,-1) b) (1,0,3) c) (-1, 0, 1)
d) (-1,1,2) e) (0,0,0) f) (1,1,1)

I have used rref(A) and think that the matrix A is not linearly independent, I think that would mean that all points can be created in R^3 so all the vectors are in the column space of matrix A.

rref(A):
(1 0 -1)
(0 1 2)
(0 0 0)

Any help of how I should think would be great.

 

[Steven G]

I have been trying to understand this question and solve it for several days. Yet i'm not getting any wiser. The question is getting harder and harder each time i'm trying to understand it. I'm a high school student.

Question; Which of the following vectors Belong to the column space of matrix A
A(3x3):
(0 1 2)
(1 1 1)
(-1 2 5)

a) (0,1,-1) b) (1,0,3) c) (-1, 0, 1)
d) (-1,1,2) e) (0,0,0) f) (1,1,1)

I have used rref(A) and think that the matrix A is not linearly independent, I think that would mean that all points can be created in R^3 so all the vectors are in the column space of matrix A.

rref(A):
(1 0 -1)
(0 1 2)
(0 0 0)

Any help of how I should think would be great.

I am impressed that you are studying Linear Algebra in high school. Since you are in HS and having trouble with this problem I will take it easy on your.
What does column space mean? How does a vector come to be in a column space?
Which of the vectors you are testing can be written as a L-C of the column vectors in A?

For the record, you are onto something if you can show that the 3 column vectors are lin dep. If that is the case then the column vectors span R^3 and all the test vectors are in the column space.

There is nothing wrong, in my opinion, that you get more and more confused as you look continue to look at a problem. Just be sure in the end you sort things out.