Hello, could anyone help me with this excersise of linear algebra, Please?
Well it's about linear transformation defined as T:V--W , the excersices give me a set of linearly independent vectors of V {v1,v2, v3... vk} and then asks :
Are {W1, w2, w3, ...wk}, the linear transformation of v1, v2,.. , also linearly independent?
How can I demonstrate that?
google the topic. You will find many good discussions, such as this one:
http://math.stackexchange.com/questions/584132/proof-of-linear-independence-of-vectors-after-applying-a-linear-transformation
Sure, I can help you with that!
To demonstrate whether the set of vectors {W1, w2, w3, ...wk} is linearly independent, we need to show that the only solution to the equation c1W1 + c2w2 + c3w3 + ... + ckwk = 0, where c1, c2, c3, ..., ck are scalars, is the trivial solution c1 = c2 = c3 = ... = ck = 0.
Here's how you can proceed:
1. Start with the equation c1W1 + c2w2 + c3w3 + ... + ckwk = 0.
2. Apply the linear transformation T to both sides of the equation. Since T is a linear transformation, we have T(c1W1 + c2w2 + c3w3 + ... + ckwk) = T(0).
3. Use the linearity property of T to rewrite the left side of the equation as c1T(W1) + c2T(w2) + c3T(w3) + ... + ckT(wk) = 0.
4. Notice that T(W1), T(w2), T(w3), ..., T(wk) are the images of the vectors W1, w2, w3, ..., wk under the linear transformation T. Let's call them v1, v2, v3, ..., vk. So the equation becomes c1v1 + c2v2 + c3v3 + ... + ckvk = 0.
5. Since the set of vectors {v1, v2, v3, ..., vk} is the linear transformation of the linearly independent set {v1, v2, v3, ..., vk}, we know that these vectors are linearly independent if and only if the only solution to the equation c1v1 + c2v2 + c3v3 + ... + ckvk = 0 is the trivial solution c1 = c2 = c3 = ... = ck = 0.
6. Therefore, to demonstrate that {W1, w2, w3, ..., wk} is linearly independent, you need to show that the only solution to c1v1 + c2v2 + c3v3 + ... + ckvk = 0 is the trivial solution c1 = c2 = c3 = ... = ck = 0.
You can approach this by using techniques such as row reduction, Gaussian elimination, or the determinant test to solve the equation. If the only solution is the trivial solution, then you can conclude that {W1, w2, w3, ..., wk} is linearly independent.
I hope this explanation helps you understand how to approach the problem! Let me know if you have any further questions.