Let \(V\) denote the set of ordered triples \((x, y, z)\) and define addition in \(V\) as in \(\RR ^3\). For each of the following definitions of scalar multiplication, select all properties of scalar multiplication for vector spaces that hold and decide whether \(V\) is a vector space.
(a)
\[a(x,y,z)=(ax, y, az)\]
Select all properties of scalar multiplication that hold true.
Closure under scalar multiplication Distributive Property over Vector Addition: \(k(\vec {u}+\vec {v})=k\vec {u}+k\vec {v}\) Distributive Property over Scalar Addition: \((k+p)\vec {u}=k\vec {u}+p\vec {u}\) Associative Property for Scalar Multiplication: \(k(p\vec {u})=(kp)\vec {u}\) Multiplication by \(1\): \(1\vec {u}=\vec {u}\)

Is \(V\) a vector space?

\(V\) is a vector space. \(V\) is not a vector space.
(b)
\[a(x, y, z)=(ax, 0, az)\]

Select all properties of scalar multiplication that hold true.

Closure under scalar multiplication Distributive Property over Vector Addition: \(k(\vec {u}+\vec {v})=k\vec {u}+k\vec {v}\) Distributive Property over Scalar Addition: \((k+p)\vec {u}=k\vec {u}+p\vec {u}\) Associative Property for Scalar Multiplication: \(k(p\vec {u})=(kp)\vec {u}\) Multiplication by \(1\): \(1\vec {u}=\vec {u}\)

Is \(V\) a vector space?

\(V\) is a vector space. \(V\) is not a vector space.
(c)
\[a(x,y,z)=(0,0,0)\]

Select all properties of scalar multiplication that hold true.

Closure under scalar multiplication Distributive Property over Vector Addition: \(k(\vec {u}+\vec {v})=k\vec {u}+k\vec {v}\) Distributive Property over Scalar Addition: \((k+p)\vec {u}=k\vec {u}+p\vec {u}\) Associative Property for Scalar Multiplication: \(k(p\vec {u})=(kp)\vec {u}\) Multiplication by \(1\): \(1\vec {u}=\vec {u}\)

Is \(V\) a vector space?

\(V\) is a vector space. \(V\) is not a vector space.
(d)
\[a(x,y,z)=(2ax, 2ay, 2az)\]

Select all properties of scalar multiplication that hold true.

Closure under scalar multiplication Distributive Property over Vector Addition: \(k(\vec {u}+\vec {v})=k\vec {u}+k\vec {v}\) Distributive Property over Scalar Addition: \((k+p)\vec {u}=k\vec {u}+p\vec {u}\) Associative Property for Scalar Multiplication: \(k(p\vec {u})=(kp)\vec {u}\) Multiplication by \(1\): \(1\vec {u}=\vec {u}\)

Is \(V\) a vector space?

\(V\) is a vector space. \(V\) is not a vector space.

Source

[Nicholson] W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2021, Open Edition, Exercises 6.1.1.