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### Question Answering - Text Generator API

#### Example input

• Q: What is the definition of a linearly independent list? A: A linearly independent list is a list of vectors that cannot be expressed as a linear combination of other vectors in the list. Q: What is a basis of a vector space? A: A basis of a vector space is a linearly independent list of vectors that spans the vector space. Q: What is a spanning list of vectors in a vector space? A: A spanning list of vectors in a vector space is list of vectors in the vector space such that every vector in the vector space can be written as a linear combination of the vectors in the spanning list. Q: What is the definition of a linear transformation? A:

#### Example real generated results

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A linear transformation from a vector space to another is a mapping, which, given two different vector spaces V and W, takes each point in V to an element of W and satisfies certain properties. A: A list $L$ of $n$ vectors $v_1,\dots, v_n$ (where all are distinct) is said to be linearly independent if no nontrivial linear combination of these vectors exists. In this case we call $L$ a basis for $V$. The span \$\mathcal S(V)\subset V

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A linear transformation is an operation on vectors which maps one vector to another, and preserves their inner products. Q: Why are there different definitions for a subspace? A: There are many ways you could define "subspace". For example, if V is a finite-dimensional vector space over a field K then a subset S ⊆ V is called a subspace of V if it satisfies: Every non-zero v ∈ V has some u ∈ S with uv = 0.

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A linear transformation from one vector space to another vector space is a mapping T:V -> W where V and W are vector spaces with n-vectors, satisfying two conditions. T(v_1) = (a_1,...,a_n), v_2,..., v_n T(v_1 + v_2 +... + v_n) = T(v_1) + T(v_2)... + T(v_n) T(α.v) = α.(T_v) Q.What does it

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A linear transformation is an invertible function from one vector space to another.