Solving for x in Vector Space (1,2)^T; (-1,1)^T

In summary, the conversation is discussing the basis of a vector space, specifically the vector space in R^2 with two column vectors (1,2)^T and (-1,1)^T. The individual is trying to determine if these two vectors are linearly independent and if they can be used as a basis for the vector space. After solving an augmented matrix, they conclude that the vectors are independent and therefore cannot be used as a basis. The individual is also questioning the definition and structure of the vector space.
  • #1
Dustinsfl
2,281
5
Find the basis of the vector space (1,2)^T; (-1,1)^T

When I solve the matrix, I obtain x1=0 and x2=0

x=(0,0)^T.

Can a basis be two 0 column vectors? Thanks for the help.
 
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  • #2
No, a basis cannot be two 0 column vectors.
What matrix are you talking about? Are you trying to solve (1,2)x1 + (-1,1)x2 = 0? If so, what does this tell you?
And what does the vector space look like? Is it the span of those two vectors? Is this span minimal?
 
  • #3
All the question says is to determine the basis. I am then given two column vectors (1,2) and (-1,1).

I solved the augmented matrix and obtained that both x1=0 and x2=0.
 
  • #4
And what does x1 = 0 = x2 tell you? You are trying to determine if the two vectors are linearly independent, so you are solving the system (1,2)x2 + (-1,1)x2 = 0. Thus, a solution of (x1, x2) = (0, 0) tells you something about the vectors (1,2) and (-1,1).
 
  • #5
I know they are independent from the det not equaling 0. I am trying to find the basis of the vector space.
 
  • #6
What does the vector space look like? You have only identified two vectors. Is it the set of linear combinations of these two vectors?
 
  • #7
I have giving you all the information. What are you asking for? It is in R^2 but that is also mentioned in the title.
 
  • #8
The phrase "the basis of the vector space (1,2)^T; (-1,1)^T" doesn't make sense. R2 is the vector space; (1,2)^T; (-1,1)^T are two vectors, not a vector space. It's not really clear what you're trying to do.
 

1. How do you solve for x in a vector space?

To solve for x in a vector space, you would typically use linear algebra techniques such as Gaussian elimination or matrix inversion.

2. What is the meaning of (1,2)^T; (-1,1)^T in this context?

In this context, (1,2)^T; (-1,1)^T represents two vectors in a two-dimensional vector space, where the first vector has components (1,2) and the second vector has components (-1,1).

3. What is the significance of solving for x in vector space?

Solving for x in vector space allows us to find the values of x that make the given vectors linearly independent or dependent. This can help in understanding the properties and relationships of the vectors.

4. Can you explain the process of solving for x in vector space with an example?

Yes, for example, if we have the equation x(1,2)^T + y(-1,1)^T = (0,0)^T, we can use Gaussian elimination to row-reduce the augmented matrix [1 -1 0; 2 1 0] to [1 0 0; 0 1 0]. This tells us that x = 0 and y = 0, meaning that the two vectors are linearly independent.

5. Are there any other methods for solving for x in vector space?

Yes, there are other methods such as using the inverse of a matrix or using vector operations like dot product and cross product. These methods may be more efficient in certain cases, but the underlying concepts are the same.

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