Simple basis / spanning set question (T / F)

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In summary, the conversation discusses the dimension of the image of a linear map T: R4 --> R7 with a kernel whose basis is v = [1 0 1 2]T. The kernel of T is the set of all vectors X in R4 that satisfy TX = 0, while the image of T is the set of all vectors Y in R7 that satisfy Y = TX. The dimension of the kernel is 1, and the dimension of the image can be determined using the rank-nullity theorem.
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theRukus
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Homework Statement


Let T: R4 --> R7 be a linear map whose kernel has the basis of v = [1 0 1 2]T. What is the dimension of the image of T?

The Attempt at a Solution


I have a very loose understanding of kernel and image, and am trying very hard to get this question. From my understanding, the kernel of T (the one in this problem) is the set of all vectors X in R4 that satisfy TX = 0. The image of T is the set of all vectors Y in R7 that satisfy Y = TX.So, from that understanding (which is very little):

Since the basis for the kernel only has one vector, X = [1 0 1 2]T (or any multiple of it). So.. The kernel has a dimension of 1.. (correct?) I'm completely unsure as to where to go from here..

Any help is greatly appreciated. Thank you.

** Sorry, I just wanted to note that I know I titled the post wrong. I was going to post a different question, but figured it out..
 
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1. Is every simple basis also a spanning set?

Yes, every simple basis is also a spanning set. A simple basis is a set of vectors that are linearly independent and span the entire vector space. This means that any vector in the vector space can be written as a linear combination of the vectors in the simple basis, making it a spanning set.

2. Can a spanning set also be a simple basis?

No, a spanning set cannot always be a simple basis. A spanning set is a set of vectors that can span the entire vector space, but it may contain linearly dependent vectors. A simple basis, on the other hand, must consist of linearly independent vectors.

3. Is a simple basis unique?

Yes, a simple basis is unique. This means that there is only one possible simple basis for a given vector space. However, a vector space may have multiple spanning sets that can be used to represent it.

4. How do you determine if a set of vectors is a simple basis?

To determine if a set of vectors is a simple basis, you can perform the following steps:

  1. Check if the vectors are linearly independent by setting up a system of equations and solving for the coefficients. If the only solution is the trivial solution (all coefficients are 0), then the vectors are linearly independent.
  2. Check if the vectors span the entire vector space by verifying if every vector in the vector space can be written as a linear combination of the vectors in the set.
  3. If both conditions are met, then the set of vectors is a simple basis.

5. Can a simple basis have more vectors than the dimension of the vector space?

No, a simple basis cannot have more vectors than the dimension of the vector space. The dimension of a vector space is defined as the number of vectors in a basis for that vector space. Since a simple basis is a basis, it must have the same number of vectors as the dimension of the vector space.

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