Linear Algebra Proof Homework: Prove p < or = 10, Linearly Independent Columns

In summary, the conversation discusses a 2000 x 10 matrix A and p vectors v1, v2, ..., vp in R10, where Av1, Av2, ..., Avp are linearly independent in R2000. The task is to prove that p is less than or equal to 10 and to show that if p = 10, then the columns of A are linearly independent. The attempt at a solution considers the relationship between the products Av1, Av2, ..., Avp and the 10 unknowns, and questions if the linear combination of these products can be linearly dependent.
  • #1
sweetiepi
23
0

Homework Statement



Let A be a 2000 x 10 matrix and v1, v2, ... , vp vectors in R10. Suppose that Av1, Av2, ... , Avp are linearly independent vectors in R2000.

a) Prove that p is < or = to 10
b) Prove that if p = 10, the columns of A are linearly independent


Homework Equations



Given above

The Attempt at a Solution



At first my line of thinking was that the products Av1, Av2 etc each had 10 unknowns and that these were somehow all related so that if there were more than 10 terms of Av1, Av2, ... , Avp then the linear combination would be linearly dependent. But I think at this point I'm just confusing myself, and it's difficult for me to picture a linear combination of a linear combination... So any help would be greatly appreciated!
 
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  • #2
If [tex]\{Av_1, \dots, Av_p\}[/tex] is linearly independent, can [tex]\{v_1, \dots, v_p\}[/tex] be linearly dependent?
 

Related to Linear Algebra Proof Homework: Prove p < or = 10, Linearly Independent Columns

1. What is a linear algebra proof?

A linear algebra proof is a mathematical method of providing evidence or justification for a statement or proposition using concepts and techniques from linear algebra. It typically involves showing that a given set of equations or properties hold true for a specific case or set of cases.

2. What does it mean for columns to be linearly independent?

Linear independence refers to a set of vectors in a vector space, where no vector in the set can be expressed as a linear combination of the other vectors. In other words, no vector in the set is redundant or can be "created" using a combination of the other vectors.

3. How do you prove that p ≤ 10?

To prove that p ≤ 10, we need to show that the maximum possible value of p is 10. This can be done by considering all possible cases and showing that in each case, p cannot exceed 10. Alternatively, we can use mathematical techniques such as induction or contradiction to prove this statement.

4. Why is it important for columns to be linearly independent?

Linear independence is important because it allows us to solve systems of equations and perform other operations on vectors and matrices. If columns are not linearly independent, it means that there is redundant information in the data, which can lead to incorrect or inconsistent results.

5. Can columns be linearly dependent while p ≤ 10?

Yes, columns can be linearly dependent while p ≤ 10. This means that although the columns are not independent, the maximum value of p is still 10. However, it is also possible for columns to be linearly dependent and for p to exceed 10, which would make the statement "p ≤ 10" false.

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