Question

Use the inner product axioms and other results of this section to verify the statements in Exercises 15–18.

15.\(\left\langle {{\rm{u,}}\,c{\rm{v}}} \right\rangle = c\left\langle {{\rm{u,}}\,{\rm{v}}} \right\rangle \) for all scalars \(c\).

Short answer

The statement \(\left\langle {{\rm{u}},c{\rm{v}}} \right\rangle = c\left\langle {{\rm{u}},{\rm{v}}} \right\rangle \) is verified.

Step-by-step solution

Apply axiom 1

According to axiom 1, if\({\bf{u}}\)and\({\rm{v}}\)bepair of vectors in a vector space\(V\), then, the inner product on\(V\),relates a real number\(\left\langle {{\bf{u}},{\rm{v}}} \right\rangle \)and satisfies the axiom,\(\left\langle {{\bf{u}},{\rm{v}}} \right\rangle = \left\langle {{\rm{v}},{\rm{u}}} \right\rangle \)for all \({\bf{u}}\),\({\rm{v}}\),\({\rm{w}}\)and scalars\(c\).

Apply axiom 1 to the left side of the given equation, as follows:

\(\left\langle {{\rm{u}},c{\rm{v}}} \right\rangle = \left\langle {c{\rm{v}},{\rm{u}}} \right\rangle \)

Apply axiom 3

According to axiom 3, if\({\bf{u}}\)and\({\rm{v}}\)bepair of vectors in a vector space\(V\), then, the inner product on\(V\),relates a real number\(\left\langle {{\bf{u}},{\rm{v}}} \right\rangle \)and satisfies the axiom,\(\left\langle {c{\rm{u}},{\rm{v}}} \right\rangle = c\left\langle {{\rm{u}},{\rm{v}}} \right\rangle \)for all \({\bf{u}}\),\({\rm{v}}\),\({\rm{w}}\)and scalars\(c\).

Apply axiom 3 to the resulting equation, as follows:

\(\begin{aligned}\left\langle {{\rm{u}},c{\rm{v}}} \right\rangle = \left\langle {c{\rm{v}},{\rm{u}}} \right\rangle \\ = c\left\langle {{\rm{v}},{\rm{u}}} \right\rangle \end{aligned}\)

Again, apply axiom 1 to the resulting equation:

\(\begin{aligned}\left\langle {{\rm{u}},c{\rm{v}}} \right\rangle = c\left\langle {{\rm{v}},{\rm{u}}} \right\rangle \\ = c\left\langle {{\rm{u}},{\rm{v}}} \right\rangle \end{aligned}\)

Thus, the statement \(\left\langle {{\rm{u}},c{\rm{v}}} \right\rangle = c\left\langle {{\rm{u}},{\rm{v}}} \right\rangle \) is verified.