Conceptual Assignment


Question Description

Recent Concepts:

For this assignment, you will write a document explaining the connections between the following course concepts:

. Spanning set of a vector space

. Linear dependence / independence of a set of vectors

. Basis of a vector space

. Dimension of a vector space

. The 4 subspaces of a matrix

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Conceptual Assignment



Conceptual Assignment
Spanning set of a vector space

A vector is called a spanning set in linear algebra if the smallest subspace that contains the
set is the linear span of a set S. assume that the subset of vector space V is S={v1,v2,}. The
case can be said that S spawns V because every vector contained in the subspace V can be written
in a linear combination of S vectors. The span of a set therefore refers to the set of all linear
combinations of the vectors in the subset, in this case vectors in S if S= {v1,v2,} is set of
vectors in subspace V (Strang, 2008).
What I learn from this relationship is that I can use a subset of the vector space of a vector
to represent all vectors. In this case, I can use the smallest part of V, which is S, to represent all
vectors that are in subspace V. below is the form taken by a linear combination created by vectors
and scalars in the subset S of V. the combination takes; a=k1v1+k2v2+k3v3+...+knvn where K and
V represents the scalars and vectors from the subset respectively (Strang, 2008).
In this case, the vectors that can be reached when S is the subset of vector space V can be
created using this combination. We used (a) to represent the span or the linear combination. The
theorem of the linear relationship however sets that the spann (S) must be a subspace of V as far
as S is a set of vectors in vector space V.
In theoretical terms, the spann (S) must be present at every subspace of V that contains S
since the spann (S) is the smallest subspace that contains vector S= {v1,v2,} in the vector V.
The simplest way to know if a set of vectors spawns a space is by using the Gaussian elimination
and checking whether there are three non-zero rows at the end (Strang, 2008).


Linear dependence/independence set of vectors

In the previous section, we have discussed construction of a spanning set of vector space.
In this section, we can ask what subspace will equal to the whole vector space in the linear
relationship (Strang, 2008).
Let us assume that there are at least two vectors in...

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