dmitriylm said:Homework Statement
Let V be the subspace spanned by the following vectors:
[ 0]...[ 1 ]...[2]
[ 2]...[ 1 ]...[5]
[-1]...[3/4]...[0]
Determine a basis for V.
The Attempt at a Solution
I'm not quite sure how to start here. Would placing the vectors in a matrix and deriving its reduced echelon form give me a basis?
Mark44 said:Are these row vectors or column vectors?
A basis is a set of linearly independent vectors that span a vector space. This means that any vector in the vector space can be expressed as a linear combination of the basis vectors.
A set of vectors form a basis if they are linearly independent and span the entire vector space. This means that none of the vectors in the set can be expressed as a linear combination of the other vectors, and any vector in the vector space can be written as a linear combination of the basis vectors.
To determine a basis given a set of vectors, you first need to check if the vectors are linearly independent. If they are not, you can use Gaussian elimination to reduce the vectors to their row echelon form. The non-zero rows of the row echelon form will form the basis. If the vectors are already linearly independent, they already form a basis.
Yes, a vector space can have multiple bases. This is because there can be different combinations of linearly independent vectors that span the same vector space.
You can check your answer by confirming that the basis vectors are linearly independent and that any vector in the vector space can be written as a linear combination of the basis vectors. You can also use the basis to solve for the coefficients of a given vector to see if it can be expressed as a linear combination of the basis vectors.