Values for which a set of vectors form a basis of Rn

That is, the first two rows are independent but the third row is the sum of the first two.In summary, to determine the values of λ for which the set of vectors {(λ^2-5, 1, 0), (2, -2, 3), (2, -3, -3)} forms a basis for ℝ^3, we need to check if the set spans ℝ^3 and if the set is linearly independent. By performing row reduction on the matrix, we see that the set is linearly independent for all values of λ except when λ^2 - 5 = 0. Therefore, the set forms a basis for ℝ^3 for all values of λ except when λ
  • #1
otapia13
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Homework Statement



For what value(s) of λ is the set of vectors {(λ^2-5, 1, 0), (2, -2, 3), (2, -3, -3)} form a basis of ℝ^3

Homework Equations



in order for a vector to form a basis it has to span R3 and the set has to be linearly independent.

The Attempt at a Solution


i tried doing row reduction on the matrix but i end up with identity matrix.
which means it would be a basis for any value, which is impossible.

the matrix I'm getting is [1, 0, 0 ; 0, 1, 0; λ^2 -5, 0, 4,]anybody??
 
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  • #2
What do you mean you get the identity matrix when you then write "the matrix I'm getting is [1, 0, 0 ; 0, 1, 0; λ^2 -5, 0, 4,]"? That's not row reduced. Or, rather, it is row reduced if and only if λ^2 -5= 0.
 

FAQ: Values for which a set of vectors form a basis of Rn

What does it mean for a set of vectors to form a basis of Rn?

When a set of vectors forms a basis of Rn, it means that these vectors can be used to represent any vector in Rn. This set of vectors is also linearly independent, meaning that no vector in the set can be written as a linear combination of the others.

How can I determine if a set of vectors forms a basis of Rn?

A set of vectors forms a basis of Rn if and only if the vectors are linearly independent and span Rn. This can be checked by using row reduction to determine if the vectors are linearly independent, and then checking if the vectors span Rn by trying to write a vector in Rn as a linear combination of the given vectors.

Can a set of more than n vectors form a basis of Rn?

No, a set of more than n vectors cannot form a basis of Rn. This is because Rn has n dimensions, and a basis must have exactly n linearly independent vectors to span Rn.

Is a basis of Rn unique?

Yes, a basis of Rn is unique. This means that for any given vector space, there is only one set of vectors that can form a basis for that space. However, there can be multiple bases for a single vector space.

How is the dimension of a vector space related to its basis?

The dimension of a vector space is equal to the number of vectors in its basis. This means that the dimension of Rn is equal to the number of vectors in its basis, and any set of n linearly independent vectors in Rn can be used as a basis for that space.

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