Help with a linear algebra problem

In summary, the conversation discusses the concepts of additive functions and linear transformations between vector spaces over the field of rational numbers. It is stated that if a function T from V to W is additive, then it must be a linear transformation. The second part of the conversation involves proving that if T is a linear transformation that maps a linearly independent subset in V to a linearly independent subset in W, then it is a one-to-one function. The conversation provides a hint to assume the mapping is from a linearly independent subset in V to a linearly dependent subset in W and then states the need to prove the other direction as well.
  • #1
kant
388
0
1)

Any fuction T: V->W . V and W are vector spaces over the field of rational numbers. The fuction T is called additive if T(x+y)=T(x)+ T(y).

Proof that any function T from v to w are additive, then it must be a linear transformation.



2) Let T:V->W be linear

prove that if T is 1 to 1 IFF T carries linear indep subset in V to Linear, indep subset in W.
 
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  • #2
kant said:
1)

Any fuction T: V->W . V and W are vector spaces over the field of rational numbers. The fuction T is called additive if T(x+y)=T(x)+ T(y).

Proof that any function T from v to w are additive, then it must be a linear transformation.



2) Let T:V->W be linear

prove that if T is 1 to 1 IFF T carries linear indep subset in V to Linear, indep subset in W.

1. Hint - Write the definition of a linear transformation right next to the question.

2. Hint - Assume it maps a lin. ind. subset in V to a lin. dep. subset to W.

That's only one direction, but now you have to go the other way too. (if and only if)
 
  • #3



1) To prove that a function T from V to W is a linear transformation, we need to show that it satisfies two properties: additivity and scalar multiplication. We are given that T is additive, which means that T(x+y) = T(x) + T(y) for any vectors x and y in V. Now, let's consider scalar multiplication. Since V and W are vector spaces over the field of rational numbers, we can multiply vectors by rational numbers. Therefore, if we have a vector v in V and a rational number r, we can define the vector rv in V as rv = r*v, where * is the scalar multiplication operation. Similarly, we can define scalar multiplication in W. Now, if we multiply a vector v in V by a rational number r and apply the function T, we should get the same result as first applying T to v and then multiplying by r. In other words, T(rv) = rT(v). This is true because T is a linear transformation and satisfies the property of scalar multiplication. Therefore, we have shown that T is both additive and satisfies scalar multiplication, and thus, it is a linear transformation.

2) To prove that if T is 1 to 1, then it carries linearly independent subsets in V to linearly independent subsets in W, we can use the contrapositive statement. In other words, we can prove that if T carries linearly dependent subsets in V to linearly dependent subsets in W, then T is not 1 to 1. Let's assume that T carries a linearly dependent subset {v1, v2, ..., vn} in V to a linearly dependent subset {w1, w2, ..., wn} in W. This means that there exist scalars a1, a2, ..., an (not all zero) such that a1w1 + a2w2 + ... + anwn = 0. Now, since T is a linear transformation, we know that T(av1 + bv2 + ... + cvn) = aT(v1) + bT(v2) + ... + cT(vn) = a0 + b0 + ... + c0 = 0, where a, b, c are scalars and T(vi) = wi. This means that the set {av1 + bv2 + ... + cvn} is a linearly dependent subset in V. Since T carries a
 

Related to Help with a linear algebra problem

1. What is linear algebra and why is it important?

Linear algebra is a branch of mathematics that deals with linear equations and their representations in vector spaces. It is important because it has many applications in various fields such as physics, engineering, and computer graphics.

2. How do I solve a linear algebra problem?

To solve a linear algebra problem, you need to first understand the problem and what is being asked. Then, you can use various techniques such as Gaussian elimination, matrix operations, and eigenvalue decomposition to find a solution.

3. Can you give an example of a linear algebra problem?

One example of a linear algebra problem is finding the inverse of a matrix. This involves using matrix operations to find a matrix that, when multiplied with the original matrix, results in the identity matrix.

4. What are some common mistakes to avoid when solving a linear algebra problem?

Some common mistakes to avoid when solving a linear algebra problem include not understanding the problem fully, not checking your work for errors, and not using the correct techniques for solving the problem.

5. Are there any online resources that can help me with linear algebra problems?

Yes, there are many online resources such as tutorials, videos, and practice problems that can help you with linear algebra problems. Some popular resources include Khan Academy, MIT OpenCourseWare, and Coursera.

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