Linear Dependence of Vectors Spanning a Space: Example Needed

In summary, a set of vectors is considered a base if they span a space and are linearly independent. However, it is possible for a set of vectors to span a space but still be dependent and not form a base. A simple example of this is when one vector in the set is a multiple of another vector. For instance, the set \left \{ (1,0),(0,1),(1,2) \right \} spans $\mathbb{R}^2$ but is not a basis due to the last vector being a multiple of the first one.
  • #1
Yankel
395
0
Hello

A base of some space is a set of vectors which span the space, and are also linearly independent.

I am looking for an example of vectors which DO span some space, but are dependent and thus not a base...can anyone give me a simple example of such a case ?

thanks !
 
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  • #2
The easiest way I can think of generalizing this is to start with a basis and then just add one more vector to the set which is a multiple of one of the other vectors.

For example, the vectors \(\displaystyle \left \{ (1,0),(0,1),(1,2) \right \}\) span $\mathbb{R}^2$ but this set isn't a basis.
 
  • #3
exactly what I was looking for, thanks !
 

FAQ: Linear Dependence of Vectors Spanning a Space: Example Needed

What is linear dependence?

Linear dependence is a mathematical concept that refers to the relationship between vectors. It means that one vector can be expressed as a combination of other vectors in a linear fashion.

What does it mean for vectors to span a space?

When a set of vectors spans a space, it means that they can be used to create any vector within that space through linear combinations. In other words, they can reach every point in that space.

Can you provide an example of linear dependence of vectors spanning a space?

Let's say we have two vectors, v1 = [1,3,2] and v2 = [2,6,4]. These two vectors are linearly dependent because v2 is simply v1 multiplied by 2. Therefore, any vector in the span of v1 and v2 can be expressed as a linear combination of these two vectors.

How can you check for linear dependence of vectors?

To check for linear dependence, you can use the determinant of the matrix formed by the vectors. If the determinant is equal to 0, then the vectors are linearly dependent. Another method is to try and find a non-trivial linear combination of the vectors that equals 0. If such a combination exists, the vectors are linearly dependent.

Why is understanding linear dependence of vectors spanning a space important?

Linear dependence is important because it helps us understand the relationships between vectors and how they can be used to represent and reach different points in a space. It also has practical applications in fields such as physics, engineering, and computer science.

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