Linear Independence and Spanning

In summary, linear independence refers to a set of vectors in a vector space that cannot be written as a linear combination of other vectors. To determine if a set of vectors is linearly independent, the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0 must be when all coefficients are equal to 0. Spanning, on the other hand, refers to the ability of a set of vectors to create all possible vectors in a vector space through linear combinations. To determine if a set of vectors spans a vector space, it must be checked if the vectors are linearly independent and if there are enough vectors to create all possible combinations. It is possible for a set of vectors to be both
  • #1
Jack Nagel
7
0
Say that {W1, W2, W3, W4} is linearly independent in R4.

Now say I have this vector
[ 2
tan(h)
7
4sec(k)
]

and I want to find values of h and k such that it is not in the span of (W1...W4).

If I understand this correctly, it means it is impossible to find those values since they do not exist. If a set of 4 vectors in R4 are linearly independent, then they also span R4.

Am I right or way off on this?
 
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  • #2
That sounds right.
 

FAQ: Linear Independence and Spanning

1. What is linear independence?

Linear independence refers to a set of vectors in a vector space that cannot be written as a linear combination of other vectors in the same space. In other words, the vectors are unique and cannot be duplicated or created by combining other vectors.

2. How do you determine if a set of vectors is linearly independent?

A set of vectors is linearly independent if the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0 is when all the coefficients (c1, c2, ..., cn) are equal to 0. In other words, the only way to get a linear combination of 0 is by multiplying all the vectors by 0.

3. What is spanning?

Spanning refers to the ability of a set of vectors to create all possible vectors in a vector space through linear combinations. If a set of vectors spans a vector space, then any vector in that space can be written as a linear combination of those vectors.

4. How do you know if a set of vectors spans a vector space?

A set of vectors spans a vector space if every vector in that space can be written as a linear combination of the given vectors. This can be determined by checking if the vectors are linearly independent and if there are enough vectors to create all possible combinations.

5. Can a set of vectors be both linearly independent and spanning?

Yes, a set of vectors can be both linearly independent and spanning. This means that the vectors are unique and cannot be created by combining other vectors, but they can still create all possible vectors in the vector space through linear combinations.

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