Finding a basis for the range space

In summary, the conversation is about finding the dimension of V and determining the basis for the column space of a matrix. It is mentioned that the basis can be written without verifying linear independence, but it is pointed out that the three vectors in the basis are actually linearly dependent. It is suggested to choose three other columns to get an independent set. The speaker then states that they already know how to solve the problem.
  • #1
Janiceleong26
276
4
1. Homework Statement
image.jpg


I've found the dimension of V to be 3.
According to the solutions, it seems that the basis can be written straight away, { (1,1,1,2), (1,2,-3,1), (3,4,-1,5) } (which is also the basis for the column space of the matrix), without verifying the vectors are linearly independent.. how come? The vectors in the matrix are not necessary linearly independent..
 
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  • #2
Those three vectors cannot be a basis of anything, because they are linearly dependent. You can choose any three other columns to get an independent set.
 
  • #3
mfb said:
Those three vectors cannot be a basis of anything, because they are linearly dependent. You can choose any three other columns to get an independent set.
Oh right.. I know how to do already, thanks!
 

1. What is a basis for the range space?

A basis for the range space is a set of linearly independent vectors that span the space of all possible outputs of a linear transformation. In other words, it is a set of vectors that can be combined in different ways to represent any output of the linear transformation.

2. How do I find a basis for the range space?

To find a basis for the range space, you can use the following steps:

  1. Apply the linear transformation to a set of linearly independent vectors.
  2. Collect the resulting vectors and determine if they are linearly independent.
  3. If they are linearly independent, they form a basis for the range space.
  4. If they are not linearly independent, use the Gram-Schmidt process to orthogonalize the vectors and then check for linear independence again.

3. Why is finding a basis for the range space important?

Finding a basis for the range space is important because it allows us to understand the behavior of a linear transformation and its output. It also helps us to determine the dimension of the range space, which is a fundamental property of a linear transformation.

4. Can there be more than one basis for the range space?

Yes, there can be more than one basis for the range space. This is because a basis is not unique and there can be different sets of linearly independent vectors that span the same space. However, all bases for the range space will have the same number of vectors, which is known as the dimension of the range space.

5. How does finding a basis for the range space relate to the null space?

Finding a basis for the range space and the null space are related because they both provide information about the behavior of a linear transformation. The basis for the range space tells us about the outputs of the transformation, while the basis for the null space tells us about the inputs that result in the zero vector as an output. Together, they give us a complete understanding of the linear transformation.

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