The notion of vector space and linear combination of vectors

Vectors: The notion of vector and vector space

The notion of vector space and linear combination of vectors

The sets \(\mathbb{R}^2\) and \(\mathbb{R}^3\), together with the componentwise addition and scalar multiplication, are examples of what in mathematics is called a vector space. These sets have the following properties of a vector space.

Definition of a vector space A nonempty set \(V\) equipped with an addition and scalar multiplication is called a vector space if the following properties hold for all scalars \(\lambda\) and \(\mu\), and for all \(\vec{u}, \vec{v}, \vec{w} \in V\):

Commutativity: \(\vec{u}+\vec{v}=\vec{v}+\vec{u}\).
Associativity: \((\vec{u}+\vec{v})+\vec{w}=\vec{u}+(\vec{v}+\vec{w})\).
Zero vector: there is an element \(\vec{0}\) in \(V\) such that \(\vec{v}+\vec{0}=\vec{v}\).
Opposite vector: every \(v\in V\) has an opposite vector \(-\vec{v}\) such that \(\vec{v}+(-\vec{v})=\vec{0}\).
Scalar 1: \(1\cdot \vec{v}=\vec{v}\).
Associativity of scalar multiplication: \(\lambda\cdot(\mu\cdot \vec{v})=(\lambda\cdot\mu)\cdot \vec{v}\).
Distributivity of scalar multiplication over vector addition: \(\lambda\cdot (\vec{u}+\vec{v})=\lambda\cdot\vec{u}+\lambda\cdot\vec{v}\).
Distributivity of scalar multiplication over scalar addition: \((\lambda+\mu)\cdot \vec{v}=\lambda\cdot\vec{v}+\mu\cdot\vec{v}\).

All properties can be proved for \(\mathbb{R}^2\) and \(\mathbb{R}^3\) by expansion of the expressions in terms of components and application of the componentwise definitions of addition and scalar multiplication.

The set of number may be the set of complex numbers instead of the real numbers. Addition and scalar multiplication in \(\mathbb{C}^2\) and \(\mathbb{C}^3\) can be defined componentwise: we only have to consider the components of a vector or a scalar to be a complex number.

A nontrivial example of a vector space is the set of quadratic functions with the usual addition of functions, and multiplication of a function with a real number. Here, one must consifer constant functions and linear functions as quadratic functions with particular coefficients equal to \(0\).

More generally, we can consider the set \(\mathcal{P}_n\) of polynomials in one variable, say \(x\), with degree at most equal to \(n\) as a vector space with the usual addition and scalar multiplication. This means that we define the vector addition and scalar multiplications as follows: let \[ \begin{array}{rcl}
a(x) &=&a_n x^n +a_{n-1}x^{n-1} +\cdots +a_1 x +a_0 \\
b(x) &=& b_n x^n +b_{n-1}x^{n-1} +\cdots +b_1 x +b_0
\end{array} \] be two polynomials in \(\mathcal{P}_n\) (note that \(a_n\) and/or \(b_n\) may be equal to \(0\)) and let \(\lambda\) be a real or complex number, then we define \[\begin{array}{rl}
a(x)+b (x)&= (a_n +b_n )x^n +(a_{n-1} +b_{n-1})x^{n-1} +\cdots +(a_1 +b_1 )x +(a_0 +b_0 ) \\
\lambda \cdot a(x) &=\lambda\cdot a_n x^n +\lambda\cdot a_{n-1}x^{n-1} +\cdots +\lambda\cdot a_1 x +\lambda\cdot a_0
\end{array} \]

Also the set of all polynomials in one variable can be considered as a vector space. For each 2-tuple of polynomials we can give them the same degree by setting enough coefficients equal to zero.

The properties above also make computational work in vector spaces easy: for example, according to the commutativity and associativity of addition of vectors it does not matter in what order addition is done. Scalar multiplication and addition of vectors may be combined at one's will.

Linear combination of vectors

A vector \(\vec{v}\) is a linear combination of vectors \(\vec{v}_1\), \(\vec{v}_2, \ldots , \vec{v}_n\) if there exist scalars \(\lambda_1\), \(\lambda_2, \ldots , \lambda_n\) such that \[\vec{v}=\lambda_1 \cdot\vec{v}_1 + \lambda_2\cdot \vec{v}_2 + \cdots + \lambda_n\cdot \vec{v}_n\] The scalars \(\lambda_1\), \(\lambda_2, \ldots , \lambda_n\) are called the coefficients of the linear combination. We then also say that the vector \(\vec{v}\) linearly depends on the vectors \(\vec{v}_1\), \(\vec{v}_2, \ldots , \vec{v}_n\). A linear combination is called nontrivial if at least one of its coefficients is nonzero. The vectors \(\vec{v}_1\), \(\vec{v}_2, \ldots , \vec{v}_n\) are linearly independent if one cannot construct a nontrivial linear combination with them equal to the zero vector.

With the term 'linear combination' we actually give a name to vectors that we can build from a given set of vectors using the two operations vector addition and scalar multiplication.

Compute the following linear combination of vectors.

Give your answer in the form of a list \([x,y]\) or vector \(\cv{x\\y}\), in which \(x\) and \(y\) are the components of the result.

\(-\cv{-3\\ 7}+2\cv{1\\ -1}={}\)\(\cv{5\\ -9}\)

After all: \[\begin{aligned}
-\cv{-3\\ 7}+2\cv{1\\ -1}
&=\cv{-1 \cdot -3 \\-1 \cdot 7} + \cv{2 \cdot 1 \\ 2 \cdot -1}
&\blue{\text{multiplication by a scalar}} \\[0.25cm]
&=\cv{3\\-7} + \cv{2\\-2}
&\blue{\text{simplification}} \\[0.25cm]
&=\cv{3 + 2\\-7 -2}
&\blue{\text{componentwise addition}} \\[0.25cm]
&= \cv{5\\ -9} &\blue{\text{final result}}
\end{aligned}\]

New example

Linear span Let \(n\) be a natural number and let \(\vec{v}_1 ,\ldots ,\vec{v}_n\) be \(n\) vectors in a vector space.

The set of all linear combinations of \(\vec{v}_1, \ldots, \vec{v}_n\) is called the space spanned by the vectors \(\vec{v}_1, \ldots, \vec{v}_n\) or their (linear) span, and is denoted as \[\sbspmatrix{ \vec{v}_1 , \ldots, \vec{v}_n}\tiny.\]

By convention, the linear span of nothing (the empty set) is equal to the span of the zero vector: \(\sbspmatrix{}=\sbspmatrix{\vec{0}}=\{\vec{0}\}\).

If \(\{\vec{v}_1,\ldots, \vec{v}_n\}\) is the smallest set of vectors that spans the vector space \(\sbspmatrix{ \vec{v}_1 , \ldots, \vec{v}_n}\), in other words, if the vectors \(\vec{v}_1,\ldots, \vec{v}_n\) are linearly independent, then \(\{\vec{v}_1,\ldots, \vec{v}_n\}\) is called a basis of the linear span and \(n\) is the dimension of this spanned vector space.

The zero vector always belongs to the linear span of vectors: it is the linear combination of \(\vec{v}_1, \ldots, \vec{v}_n\) where all coefficients \(\lambda_i\) are equal to \(0\).

The linear span of a single vector \(\vec{v}\) in \(\mathbb{R}^2\) or \(\mathbb{R}^3\) is a straight line through the origin \(\vec{0}\) and \(\vec{v}\). The parametric representation of this line is \(\lambda\cdot \vec{v}\)

The linear span of two vectors \(\vec{u}\) and \(\vec{v}\) in \(\mathbb{R}^3\) which are not scalar multiples of each other, is a plane through the origin \(\vec{0}\), \(\vec{u}\) and \(\vec{v}\).

Example Polynomials form a vector space. The linear span of the powers \(1, x, x^2\) and \(x^3\) is equal to the collection of all polynomials of degree 3 or less. After all, a randomly chosen polynomial \(a_3x^3+a_2x^2+a_1x+a_0\) can be written as \(a_3\cdot x^3+a_2\cdot x^2+a_1\cdot x+a_0\cdot 1\) and this means that any polynomial of degree 3 or less belongs to the linear span \(\sbspmatrix{x^3,x^2,x,1}\). The set \(\{1,x,x^2,x^3\}\) is a basis of the vector space \(\mathcal{P}_3\). The dimension \(\mathcal{P}_3\) is equal to \(4\).