Consider the complex vector space V=ℂ^3 and the list (v1, v2, v3) of vectors in V, where v1=(i,

step 1 To show that U1, U2, and U3 is a basis, we need to show that the three vectors are linearly inde

Consider the complex vector space V=ℂ^3 and the list (v1,...

Consider the complex vector space $V=\mathbb{C}^{3}$ and the list $\left(v_{1}, v_{2}, v_{3}\right)$ of vectors in $V$, where $$v_{1}=(i, 0,0), v_{2}=(i, 1,0), v_{3}=(i, i,-1)$$ (a) Prove that $\operatorname{span}\left(v_{1}, v-2, v_{3}\right)=V$. (b) Prove or disprove: $\left(v_{1}, v_{2}, v_{3}\right)$ is a basis for $V$.

Consider the complex vector space $V=\mathbb{C}^{3}$ and the list $\left(v_{1}, v_{2}, v_{3}\right)$ of vectors in $V$, where
$$v_{1}=(i, 0,0), v_{2}=(i, 1,0), v_{3}=(i, i,-1)$$
(a) Prove that $\operatorname{span}\left(v_{1}, v-2, v_{3}\right)=V$.
(b) Prove or disprove: $\left(v_{1}, v_{2}, v_{3}\right)$ is a basis for $V$.

Key Concepts

Basis

A basis of a vector space is a set of vectors that is both linearly independent and spanning. This means every vector in the space can be uniquely expressed as a linear combination of the basis vectors. The concept of a basis is fundamental because it allows the precise characterization of the structure and dimensionality of the vector space.

Linear Independence

Vectors are said to be linearly independent if no vector in the set can be written as a linear combination of the others. This property is crucial when forming a basis because it ensures that each vector contributes a unique direction to the vector space. Linear independence also guarantees that the representation of any vector in the space using the basis is unique.

Vector Space

A vector space is a collection of objects called vectors, which can be added together and multiplied by scalars (elements from a field) in such a way that the operations satisfy certain axioms like associativity, commutativity of addition, existence of an additive identity, and distributivity of scalar multiplication. In the context of complex vector spaces, the scalars are complex numbers, and properties of these spaces govern how vectors behave under linear combinations.

Span

The span of a set of vectors is the set of all possible linear combinations of those vectors. It represents the smallest subspace that contains all the given vectors. Demonstrating that the span of a set equals the entire vector space shows that the vectors collectively are sufficient to 'reach' every vector in the space, which is key in verifying completeness of a generating set.

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