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tandoorichicken
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Is it possible for a nullspace to be spanned by only one vector? Does a statement like Nul A = Span{[itex]\vec{v}_1[/itex]} even have any meaning?
A spanning vector space is a set of vectors that can be used to create any other vector in that space through linear combinations. In other words, any vector in the space can be written as a combination of the spanning vectors.
A set of vectors spans a vector space if and only if every vector in the space can be written as a linear combination of the spanning vectors. This means that for any vector in the space, there exists a unique set of coefficients that can be multiplied with the spanning vectors to create that vector.
No, a set of vectors can only span one vector space. This is because the set of vectors must be able to create any vector within that space through linear combinations. If the set of vectors were able to create vectors in multiple spaces, it would mean that those spaces are equivalent.
Yes, it is possible for a vector space to have an infinite number of spanning vectors. This is common in vector spaces with infinite dimensions, such as the space of all real numbers or the space of all polynomials.
To find the spanning vectors for a given vector space, you can use the Gaussian elimination method. This involves creating an augmented matrix with the vectors as columns and then performing row operations to reduce the matrix to its row echelon form. The resulting vectors will be the spanning vectors for the vector space.