Definition

In linear algebra, a real number is called a scalar. A scalar can be multiplied with a vector coordinate-wise: \[x < a_1, a_2, \dots, a_n > = < xa_1, xa_2, \dots, xa_n >\]

For an example, \[ 5 <-2,1,4> = <-10,5,20> \] Similarly, you can factor a constant from a vector. \[<2, 6, -6> = 2 <1, 3, -3>\]

Vector Subtraction

Given a vector \(\overrightarrow{a} = < a_1, a_2, \dots, a_n >,\) the additive inverse of \(\overrightarrow{a}\) is \[-\overrightarrow{a} = < -a_1, -a_2, \dots, -a_n >\] Using the definition of vector addition, we can define vector subtraction as \[\overrightarrow{b}-\overrightarrow{a} = \overrightarrow{b}+(-1\cdot\overrightarrow{a})\]

It is always the case that \[\overrightarrow{a}-\overrightarrow{a} = \overrightarrow{0}\]

Properties of Scalar Multiplication

• Commutativity: Given a scalar \(x\) and a vector \(\overrightarrow{v},\) \[x\overrightarrow{v}=\overrightarrow{v}x.\]
• Distributivity of vectors over scalars: Given scalars \(x\) and \(y\) and a vector \(\overrightarrow{v},\) \[(x+y)\overrightarrow{v}=x\overrightarrow{v} + y\overrightarrow{v}.\]
• Distributivity of scalars over vectors: Given a scalar \(x\) and vectors \(\overrightarrow{v}\) and \(\overrightarrow{w},\)\[x(\overrightarrow{v}+\overrightarrow{w}) = x\overrightarrow{v}+x\overrightarrow{w}.\]
• Associativity of scalars and vectors: Given scalars \(x\) and \(y\) and vector \(\overrightarrow{v},\)\[x(y\overrightarrow{v})=(xy)\overrightarrow{v}.\]