Dimension of span of vectors

[esliii]

Given the vectors(I couldn't find the brackets to put around the numbers):

v1 = 2
-2
1
v2 = -1
2
0
v3 = 3
-2
X
v4 = -5
6
-2

where X is a number.

Find the value of X so that the dimension of the span{v1, v2, v3, v4} is three. Then decide whether {v1, v2, v3} or {v1, v2, v4} is a basis for R^3.

 

[daon2]

Create a matrix with these vectors as columns and the put the matrix into RREF. At that point, you will be able to find x so that the rank of your matrix is 3. The pivot columns in RREF of your matrix correspond to linearly independent vectors in your original matrix.

 

[esliii]

Create a matrix with these vectors as columns and the put the matrix into RREF. At that point, you will be able to find x so that the rank of your matrix is 3. The pivot columns in RREF of your matrix correspond to linearly independent vectors in your original matrix.

What I get is that if any number besides 2 is used in place of X then the determinant of the matrix will be nonzero which means it is linearly independent right? In RREF you will get a matrix with no nonzero rows meaning that the dimension of the span{v1, v2, v3, v4} is 3 right?

 

[daon2]

What I get is that if any number besides 2 is used in place of X then the determinant of the matrix will be nonzero which means it is linearly independent right? In RREF you will get a matrix with no nonzero rows meaning that the dimension of the span{v1, v2, v3, v4} is 3 right?

I assume you have an entry of 4-2x? Well, what happens if x=2? What is the rank of your matrix then? The reasoning in your last sentence is okay. However, you cannot take the determinant of a non-square matrix!

 

[esliii]

I assume you have an entry of 4-2x? Well, what happens if x=2? What is the rank of your matrix then? The reasoning in your last sentence is okay. However, you cannot take the determinant of a non-square matrix!

Not sure I understand what you mean about x=2??? Please explain! I MUST understand this!

 

[daon2]

If x=2 that vector is linearly dependent. That helps you answer the second part.

 

[esliii]

If x=2 that vector is linearly dependent. That helps you answer the second part.

Ok yeah that makes sense. Thanks. But I have one more question: Is 4-2x what you get for X when reducing the matrix? Like solving for a variable in an algebraic expression? And thank you very much by the way... You have helped a ton.

 

[daon2]

When I row reduced the matrix the third row had as a leading entry 4-2x (or 2-x, I don't remember). I'm not sure what you mean by X?