Linear Dependence/Linear Combination Question

In summary: There are now four possible answers: p= q= c= 1, p= q= c= 2, p= q= c= 3, and p= q= c= -1. So v_3 is a linear combination of v_2 and v_3 if and only if there exist non-zero values, p and q, such that [3, a]= p[1, -1]+ q[-2, 2]+ r[3, a]= [0, 0] which is equivalent to p= q= c= 0.
  • #1
Tim 1234
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S={v1, v2, v3}

v1=[1, -1], v2=[-2, 2], v3=[3, a]

a) For what value(s) a is the set S linearly dependent?
b)For what value(s) a can v3 be expressed as a linear combination of v1 and v2?

a) p=3 and m=2
3-2=1 free variable
Therefore the set has non-trivial solutions and is linearly dependent

b) I reduced the matrix to the following:

1 -2 3 0
0 0 a+3 0

Does a need to equal -3 for a linear combination to be valid?

I recognize v2=(-2)v1 - is this relevant at all?
 
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  • #2
Tim 1234 said:
S={v1, v2, v3}

v1=[1, -1], v2=[-2, 2], v3=[3, a]

a) For what value(s) a is the set S linearly dependent?
b)For what value(s) a can v3 be expressed as a linear combination of v1 and v2?

a) p=3 and m=2
3-2=1 free variable
Therefore the set has non-trivial solutions and is linearly dependent
You haven't answered the question -- "For what value(s) of a is the set linearly dependent?" Your answer should say something about a.
Tim 1234 said:
b) I reduced the matrix to the following:

1 -2 3 0
0 0 a+3 0

Does a need to equal -3 for a linear combination to be valid?

I recognize v2=(-2)v1 - is this relevant at all?
Yes, it's relevant to both question parts. Since v2 is a multiple of v1 (and vice versa), part b boils down to the question, "what values of a make v3 a scalar multiple of either v1 or v2?
 
  • #3
Tim 1234 said:
S={v1, v2, v3}

v1=[1, -1], v2=[-2, 2], v3=[3, a]

a) For what value(s) a is the set S linearly dependent?
b)For what value(s) a can v3 be expressed as a linear combination of v1 and v2?

a) p=3 and m=2
3-2=1 free variable
Therefore the set has non-trivial solutions and is linearly dependent
In what sense does a set have "solutions" at all?
What I think you meant to say, and what you should say, is "this set is linearly independent if and only if the only values of p, q, and r, that make p[1, -1]+ q[-2, 2]+ r[3, a]= [0, 0] are p= q= c= 0." That gives the two equations p- 2q+ 3r= 0 and -p+ 2q+ 3r= 0. But p= 2, q= 1, r= 0 will work. Therefore the set is not independent.

b) I reduced the matrix to the following:

1 -2 3 0
0 0 a+3 0

Does a need to equal -3 for a linear combination to be valid?

I recognize v2=(-2)v1 - is this relevant at all?
Personally, I dislike changing everything to matrices (and you shouldn't say "I reduced the matrix" when you haven't yet shown a matrix to begin with!). [itex]v_3[/itex] is a linear combination of [itex]v_2[/itex] and [itex]v_3[/itex] if and only if there exist non-zero values, p and q, such that [3, a]= p[1, -1]+ q[-2, 2] which gives the two equations p- 2q= 3 and -p+ 2q= a. What happens if you add those two equations?
 

FAQ: Linear Dependence/Linear Combination Question

What is linear dependence?

Linear dependence refers to the relationship between vectors in a vector space. It means that one or more vectors in the space can be expressed as a combination of other vectors in the space.

What is a linear combination?

A linear combination is a mathematical operation where two or more vectors are multiplied by constants and then added together. This operation is commonly used to express linear dependence between vectors.

How do you determine if a set of vectors is linearly dependent?

To determine if a set of vectors is linearly dependent, you can use the following steps:

  1. Write the vectors as columns in a matrix.
  2. Use row operations to reduce the matrix to its row-echelon form.
  3. If there is a row of zeros in the reduced matrix, then the vectors are linearly dependent.
  4. If there are no rows of zeros, but there are more columns than rows, then the vectors are also linearly dependent.

What is the difference between linear dependence and linear independence?

Linear dependence and linear independence are two opposite concepts. Linear dependence means that one or more vectors in a vector space can be expressed as a linear combination of other vectors, while linear independence means that no vector in the space can be expressed as a linear combination of the other vectors.

Why is linear dependence important in linear algebra?

Linear dependence is important in linear algebra because it helps us understand the relationships between vectors in a vector space. It also allows us to simplify and solve complex systems of linear equations by reducing them to their row-echelon form.

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