Linear Algebra: Basis and Dimension problem

In summary, given the vectors: x1=(3,-2,4), x2=(-3,2,-4) and x3=(-6,4,-8) the dimension of Span(x1,x2,x3) is 1.
  • #1
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Homework Statement



Given the vectors: x1=(3,-2,4), x2=(-3,2,-4) and x3=(-6,4,-8) , what is the dimension of Span(x1,x2,x3)

Homework Equations





The Attempt at a Solution



I know x1,x2 and x3 are Linearly dependent since its determinant is zero. There are a total of 3 vectors in the spanning set. I thought the number of dimensions would be 3. But my the back of my book says the number of dimensions is one.
 
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  • #2
x1, x2 and x3 are scalar multiples of each other (check this for yourself). Therefore, the dimension is 1.
 
  • #3
jhicks said:
x1, x2 and x3 are scalar multiples of each other (check this for yourself). Therefore, the dimension is 1.

I do not understand how x1 , x2 and x3 being scalar multiples of each other makes the number of dimensions 1.
 
  • #4
basically, x1, x2 and x3 are linearly dependent in that each can be written in terms of only one other multiplied by a scaling constant. Since the dimension of the span is how many linearly independent vectors there are (only one in this case), the dimension of the span is 1.
 
  • #5
The definition of dimension of a space is the number of vectors in a basis. A basis is any set of vectors that both spans the space and is independent. The given set is NOT a basis specifically because it in not independent- as you say, it is dependent. If a set of vectors is dependent, at least one of the vectors can be written as a linear combination of the others and can be dropped from the set. In this case, (-3, 2, -4) = (-1)(3, -2, 4) so any vector that can be written as a linear combination of the three given vectors can be written without (-3, 2, -4): {(3, -2, 4), (-6, 4, 8)} also spans the space. But that set is also dependent: (-6, 4, 8)= -2(3, -2, 4) so any vector in its span can be written as a multiple of (3, -2, 4). That means that the set {(3, -2, 4)} both spans the space and is trivially independent. That set is a basis for the space and, because it contains only one vector, the dimension of the space is 1.
 
  • #6
you said that there are two conditions in order to be a a basis one is that the set should be linearly independent and the other condition it should span the space, my question is what is the space these vectors given do they span ? why you did not concentrate on this part of the definition of basis and only on linear independence? and these vectors are in R3, why the dimension is not 3 for example ?
 

1. What is a basis in linear algebra?

A basis in linear algebra is a set of linearly independent vectors that span the entire vector space. This means that any vector in the space can be written as a linear combination of the basis vectors. A basis is important because it allows us to represent and manipulate vectors in a systematic way.

2. What is the dimension of a vector space?

The dimension of a vector space is the number of vectors in a basis for that space. It represents the minimum number of linearly independent vectors needed to span the entire space. For example, in a three-dimensional space, three linearly independent vectors are needed to form a basis, so the dimension of that space is 3.

3. How do you find the basis of a vector space?

To find the basis of a vector space, you can start by finding a set of linearly independent vectors that span the space. This can be done using techniques such as Gaussian elimination or matrix operations. Then, check that these vectors are not redundant and that they span the entire space. If they do, then they form a basis for the vector space.

4. Can a vector space have more than one basis?

Yes, a vector space can have more than one basis. This is because there can be multiple sets of linearly independent vectors that span the same vector space. However, all bases for a given vector space will have the same number of vectors, which is equal to the dimension of the space.

5. How is the concept of basis and dimension used in real-world applications?

The concept of basis and dimension is used in various real-world applications, such as data analysis, computer graphics, and physics. In data analysis, basis vectors can be used to represent features or variables in a dataset, allowing for efficient data manipulation and analysis. In computer graphics, basis vectors can be used to represent the orientation and position of objects in 3D space. In physics, the concept of basis and dimension is fundamental in understanding and solving problems related to vectors and vector spaces.

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