Linear algebra-Basis of a linear map

In summary, the conversation discusses a linear map L from R^2 to R^2, where L is not equal to the zero map, but L squared is equal to the zero map. The problem asks to show that there exists a basis {A,B} of R^2 such that L(A) = B and L(B) = O. The solution provided states that A and B must be linearly independent, and explains that if aA+bB = O, then a and b must equal to zero. The solution also mentions that L(aA+bB) = aL(A) + bL(B) due to L being a linear map.
  • #1
manuel325
16
0

Homework Statement



Let ##L: R^{2} → R^{2}## be a linear map such that ##L ≠ O## but## L^{2} = L \circ L = O.##
Show that there exists a basis {##A##, ##B##} of ##R^{2}## such that:

##L(A) = B## and ##L(B) = O.##​

The Attempt at a Solution


Here's the solution my book provides :
problem.JPG

Well I have two questions:
1.Why do they say that ##aA+bB=O##?. I mean I don't understand the solution from that point until the end (Why the solutions ##a=0## and ##b=0## are enough to prove the existence of that basis??May someone please explain??
Thanks in advance :smile:. Any help would be appreciated
 
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  • #2
The solution says IF [itex]aA+bB=0[/itex], THEN [itex]a=b=0[/itex]. That is what it means for the vectors [itex]A[/itex] and [itex]B[/itex] to be linearly independent. Vectors in a basis must be linearly independent.
 
  • #3
krome said:
The solution says IF [itex]aA+bB=0[/itex], THEN [itex]a=b=0[/itex]. That is what it means for the vectors [itex]A[/itex] and [itex]B[/itex] to be linearly independent. Vectors in a basis must be linearly independent.
Thanks but why ##O=L(aA+bB)=aL(A)##?? could you please explain what they do there, please??
 
  • #4
manuel325 said:
Thanks but why ##O=L(aA+bB)=aL(A)##?? could you please explain what they do there, please??

[itex]L[/itex] is a linear map, which means [itex]L(aA+bB) = aL(A) + bL(B)[/itex].
 
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  • #5
krome said:
[itex]L[/itex] is a linear map, which means [itex]L(aA+bB) = aL(A) + bL(B)[/itex].

Thanks :smile:
 

1. What is the definition of a basis in linear algebra?

A basis is a set of linearly independent vectors that span the entire vector space. This means that any vector in the vector space can be expressed as a unique linear combination of the basis vectors.

2. How do you determine if a set of vectors is a basis for a linear map?

To determine if a set of vectors is a basis for a linear map, you can use the pivot columns method. If the vectors form a linearly independent set and span the entire vector space, then they are a basis for the linear map.

3. Can a linear map have more than one basis?

Yes, a linear map can have multiple bases. This is because a basis is not unique and there can be infinitely many combinations of linearly independent vectors that span the vector space.

4. What is the relationship between the dimension of a vector space and the number of vectors in a basis?

The dimension of a vector space is equal to the number of vectors in a basis. This means that the number of vectors in a basis is the minimum number of vectors needed to span the entire vector space.

5. How can you use the basis of a linear map to find the matrix representation of the linear map?

The basis vectors can be used to construct the columns of the matrix representation of the linear map. The coordinates of each vector in the basis will correspond to the entries in the matrix. By using the basis vectors, the matrix representation can be easily determined.

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