Linear Independence. express each vector as a lin. combo

In summary: Another way would be to find c1 and c3 for which v1 = c1v2 + c3v3, and then use that to express v2 as a linear combination of v1 and v3. Either way, the result would be the same.
  • #1
burton95
54
0
V1 = (1,2,3,4) V2 = (0,1,0,-1) V3 = (1,3,3,3)

a) I already expressed them a linearly dependent set in R4

b) Express each vector in part (a) as a linear combination of the other two

linear combo is just {c1v1 + c2v2...cnvn} right? But I don't get where to start to prove this
 
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  • #2
burton95 said:
V1 = (1,2,3,4) V2 = (0,1,0,-1) V3 = (1,3,3,3)

a) I already expressed them a linearly dependent set in R4

b) Express each vector in part (a) as a linear combination of the other two

linear combo is just {c1v1 + c2v2...cnvn} right? But I don't get where to start to prove this

There is nothing to prove; that is just a *definition* of "linear combination". You are being asked to represent v1 as a linear combination of v2, v3, and so forth. To start: just write things down in detail: figure out what are the components of c2v2 + c3v3 for constants c2 and c3. How can that combination be equal to v1?
 
  • #3
so are you saying take v1 = c2 (v2) + c3 (v3) using the constants that I found through proving that they are linearly dependent?
 
  • #4
What Ray is saying has three parts.
1) Find constants c2 and c3 for which v1 = c2v2 + c3v3
2) Find constants c1 and c3 for which v2 = c1v1 + c3v3
3) Find constants c1 and c2 for which v3 = c1v1 + c2v2
 
  • #5
BTW, when you post a question, do not throw away the three parts of the template. They are there for a reason.
 
  • #6
Thx. I will leave the template. So I am left with 3 different equations in the form of vn = cx(vx) + cy(vy) as my final answer?
 
  • #7
burton95 said:
so are you saying take v1 = c2 (v2) + c3 (v3) using the constants that I found through proving that they are linearly dependent?

That would be one way of doing it.
 

FAQ: Linear Independence. express each vector as a lin. combo

1. What does it mean for vectors to be linearly independent?

Linear independence refers to a set of vectors that cannot be written as a linear combination of each other. In other words, no vector in the set can be expressed as a combination of the other vectors using scalar multiplication and addition. This property is important in linear algebra and is used to determine the uniqueness of a solution to a system of linear equations.

2. How can I check if a set of vectors is linearly independent?

To check if a set of vectors is linearly independent, you can use the determinant method or the row reduction method. The determinant method involves creating a matrix with the given vectors as columns and calculating the determinant. If the determinant is non-zero, the vectors are linearly independent. The row reduction method involves creating an augmented matrix and performing row operations to reduce it to echelon form. If there are no free variables, the vectors are linearly independent.

3. Can a set of two vectors be linearly independent?

Yes, a set of two vectors can be linearly independent as long as they are not scalar multiples of each other. This means that they cannot be on the same line or parallel to each other. For example, the vectors (1, 0) and (0, 1) are linearly independent, but the vectors (2, 4) and (4, 8) are not since one is a multiple of the other.

4. How do I express a vector as a linear combination of other vectors?

To express a vector as a linear combination of other vectors, you need to find the coefficients (scalar values) that when multiplied by each vector and added together, result in the given vector. This can be done by solving a system of linear equations or by using the inverse of the matrix formed by the given vectors. If the given vectors are linearly independent, there will be a unique solution to the system of equations.

5. Why is linear independence important in linear algebra?

Linear independence is important in linear algebra because it allows us to determine the uniqueness of a solution to a system of linear equations. If a set of vectors is linearly independent, it means that each vector in the set contributes a unique direction to the solution. This property is also used in determining the basis of a vector space and in solving problems related to linear transformations.

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