# Calculating Span of 3 Vectors in R^3

• Josh123
In summary, the question is about determining if three vectors span R^3 by placing them in an augmented matrix and finding the determinant. If the determinant is 0, then the vectors do not span R^3. The conversation also touches on the concepts of linear dependence/independence and forming a basis for a subspace.
Josh123

Given 3 vectors:
v1=(1,1,2)
v2=(1,0,1)
v3=(2,1,3)

To determine if they span R^3, I placed these 3 vectors in an augmented matrix. I found the determinant to be 0 which means that the 3 vectors do NOT span R^3. My question is, since v1,v2,v3 are linearly dependent, are they not supposed to span R^3?

Yes, if three vectors are independent, then they must span a three dimensional vector space- in this case R3. I am wondering about your use of "augmented" and then "determinant". "Augmenting" this matrix would give you a 4 by 3 matrix that does not have a determinant. Of course, you are correct that the determinant formed by the vectors (not augmented) has determinant 0. Now, are you sure these vectors are independent?

What you may be thinking of is this: to determine whether the vectors are independent or not, you could set up the matrix equation corresponding to saying that some linear combination: av1+ bv2+ cv2= 0 and set up the augmented matrix corresponding to that. If the row reduction of that gives a matrix having a row of all zeros then the vectors are NOT independent. What happens here?

I got a little mixed up with the linear dependence/independence, but I understand now. Thanx

I have another question though.. just want to make sure if my answer is correct:

Let S= Span {(1,1-3), (2,2,-6) , (-4,-4,12)}.

Is the basis of vectors = {(1,1,-3)(2,2,-6), (-4,-4,12)} ?

If they (1) span that subspace, and (2) are linearly independent, then yes, that is a basis of that subspace.

I have a question that kind of relates tp Josh123's questions about spanning.

Let's say we have a subspace: (2,4,7) (1,3,2)(1,1,5)

Since one of them is a linear combination of the other two, does it mean that we only need two of those vectors to form the basis?

Span{(1,3,2) (1,1,5)}

jackdamack10 said:
I have a question that kind of relates tp Josh123's questions about spanning.

Let's say we have a subspace: (2,4,7) (1,3,2)(1,1,5)

Since one of them is a linear combination of the other two, does it mean that we only need two of those vectors to form the basis?

Span{(1,3,2) (1,1,5)}

Two of those vectors will span and form the basis for a two dimensional subspace (plane) in the three dimensional space. They can be combined with appropriate multiplying coefficients to form any vector in that plane. The only way to represent a vector that is not in the plane is to have a third vector that is linearly independent of those two.

Josh123 said:
I got a little mixed up with the linear dependence/independence, but I understand now. Thanx

I have another question though.. just want to make sure if my answer is correct:

Let S= Span {(1,1-3), (2,2,-6) , (-4,-4,12)}.

Is the basis of vectors = {(1,1,-3)(2,2,-6), (-4,-4,12)} ?

If you "understand" the first problem, how could you possibly have a problem with this? Even if you don't want to do the work of setting up a matrix and row reducing, did you notice that (2,2,-6)= 2(1,1,-3) and that
(-4,-4,12)= 4(1,1,-3)?

## 1. How do I calculate the span of 3 vectors in R^3?

The span of three vectors in R^3 can be calculated by creating a matrix with the three vectors as its columns. Then, use Gaussian elimination or other matrix operations to reduce the matrix to its row-echelon form. The number of non-zero rows in the resulting matrix will be equal to the span of the three vectors.

## 2. Can the span of 3 vectors in R^3 be greater than 3?

No, the span of three vectors in R^3 cannot be greater than 3. This is because R^3 is a three-dimensional vector space, meaning that any combination of three linearly independent vectors will span the entire space. Adding a fourth vector would result in linear dependence and the span would remain at 3.

## 3. Can the span of 3 vectors in R^3 be less than 3?

Yes, the span of three vectors in R^3 can be less than 3. This occurs when the three vectors are linearly dependent, meaning that one or more of the vectors can be written as a linear combination of the other two. In this case, the span will be equal to the number of linearly independent vectors.

## 4. What is the significance of calculating the span of 3 vectors in R^3?

The span of three vectors in R^3 is important because it represents the number of dimensions that the vectors can span. This can help determine if the vectors are linearly independent or dependent, as well as their relationship to the rest of the vector space.

## 5. How does the span of 3 vectors in R^3 relate to the concept of a basis?

The span of three vectors in R^3 is related to the concept of a basis because the span represents the minimum number of vectors needed to span the entire vector space. A basis is a set of linearly independent vectors that span the entire space, and the span of these vectors will be equal to the dimension of the vector space.

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