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cahiersujet
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What would the subspace spanned by a single vector (for example) f(x)=x+1 be?
A subspace is a subset of a vector space that has the same structure and operations as the original vector space. It is closed under addition and scalar multiplication and contains the zero vector.
A vector spans a subspace if all elements in that subspace can be expressed as a linear combination of that vector. In other words, the vector can reach every point in the subspace through linear combinations.
To determine if a vector spans a subspace, you can use the span test. This involves checking if the vector is a linear combination of the basis vectors of the subspace. If it is, then the vector spans the subspace.
Yes, it is possible for multiple vectors to span the same subspace. As long as all the vectors are linearly independent, they can span the subspace.
Understanding the concept of subspace spanned by a vector is important because it allows us to visualize and manipulate vector spaces in a more efficient way. It also helps us to solve problems related to linear equations and transformations.