22.4: Vector Spaces

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A Vector Space is a set \(V\) of elements called vectors, having operations of addition and scalar multiplication defined on it that satisfy the following conditions (\(u\), \(v\), and \(w\) are arbitrary elements of \(V\), and \(c\) and \(d\) are scalars.)

Closure Axioms

The sum \(u+v\) exists and is an element of \(V\). (\(V\) is closed under addition.)
\(cu\) is an element of \(V\). (\(V\) is closed under multiplication.)

Addition Axioms

\(u+v=v+u\) (commutative property)
\(u+(v+w)=(u+v)+w\) (associative property)
There exists an element of \(V\), called a zero vector, denoted 0, such that \(u+0=u\)
For every element \(u\) of \(V\), there exists an element called a negative of \(u\), denoted \(−u\), such that \(u+(−u)=0\).

Scalar Multiplication Axioms

\(c(u+v)=cu+cv\)
\((c+d)u=cu+du\)
\(c(du)=(cd)u\)
\(1u=u\)

Definition of a basis of a vector space

A finite set of vectors \(v_1, \ldots, v_n\) is called a basis of a vector space \(V\) if the set spans \(V\) and is linearly independent. i.e. each vector in \(V\) can be expressed uniquely as a linear combination of the vectors in a basis.

Vector spaces

Do This

Let \(U\) be the set of all circles in \(R^2\) having center at the origin. Interpret the origin as being in this set, i.e., it is a circle center at the origin with radius zero. Assume \(C_1\) and \(C_2\) are elements of \(U\). Let \(C_1 + C_2\) be the circle centered at the origin, whose radius is the sum of the radii of \(C_1\) and \(C_2\). Let \(kC_1\) be the circle center at the origin, whose radius is \(|k|\) times that of \(C_1\). Determine which vector space axioms hold and which do not.

Spans:

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Let \(v\), \(v_1\), and \(v_2\) be vectors in a vector space \(V\). Let \(v\) be a linear combination of \(v_1\) and \(v_2\). If \(c_1\) and \(c_2\) are nonzero real numbers, show that \(v\) is also a linear combination of \(c_{1}v_{1}\) and \(c_{2}v_{2}\).

Do This

Let \(v_1\) and \(v_2\) span a vector space \(V\). Let \(v_3\) be any other vector in \(V\). Show that \(v_1\), \(v_2\), and \(v_3\) also span \(V\).

Linear Independent:

Consider the following matrix, which is in the reduced row echelon form.

\[\begin{split}
\left[
\begin{matrix}
1 & 0 & 0 & 7 \\
0 & 1 & 0 & 4 \\
0 & 0 & 1 & 3
\end{matrix}
\right]
\end{split} \nonumber \]

Do This

Show that the row vectors form a linearly independent set:

Do This

Is the set of nonzero row vectors of any matrix in reduced row echelon form linearly independent? Discuss in your groups and include your thoughts below.

Do This

A computer program accepts a number of vectors in \(R^3\) as input and checks to see if the vectors are linearly independent and outputs a True/False statment. Discuss in your groups, which is more likely to happen due to round-off error–that the computer states that a given set of linearly independent vectors is linearly dependent, or vice versa? Put your groups thoughts below.