Essential Concepts in Linear Algebra
Some important concepts in linear algebra.
Table of Contents
1. Linear Independent
- Linearly Independent
- A list of vectors \(v_{1},v_{2},\dots,v_{m}\) in \(V\) is called linearly independent if the only choice of \(a_{1},a_{2},\dots,a_{m} \in F\) that makes \(\sum_{i=1}^{m} a_{i}v_{i}=0\) is \(a_{1}=a_{2}=\dots=a_{m}=0\).
Let \(v_{1},v_{2},\dots,v_{m}\) be linearly independent in \(V\) and \(w \in V\), prove that
\[ \dim \mathrm{span}(v_{1}+w, v_{2}+w, \dots, v_{m}+w) \ge m-1 \]
Consider the following \(m-1\) vectors: \(v_{i}-v_{m} = (v_{i}+w)-(v_{m}+w)\),
\[ a_{1}(v_{1}-v_{m}) + a_{2}(v_{2}-v_{m}) + \dots + a_{m-1}(v_{m-1}-v_{m}) = 0 \]
Reformat the equation:
\[ a_{1}v_{1} + a_{2}v_{2} + \dots + a_{m-1}v_{m-1} - (a_{1}+a_{2}+\dots+a_{m-1})v_{m}=0 \]
Since \(v_{i}\) are linearly independent in \(V\), therefore \(a_{i}=0\). And since \(v_{i}-v_{m}\) all belongs to \(\mathrm{span}(v_{1}+w, v_{2}+w, \dots, v_{m}+w)\), therefore, this span contains at least \(m-1\) linearly independent vectors, therefore its dimension \(\ge m-1\)
2. Basis
- Basis
- A basis of \(V\) is a list of vectors in \(V\) that is linearly independent and spans \(V\).
2.1. Subspace
Suppose \(V\) is finite-dimensional and \(U\) is a subspace of \(V\), then there exists a subspace \(W\) of \(V\) such that \(V = U \oplus W\).
Suppose the basis of \(U\) is \(u_{1}, u_{2}, \dots, u_{m}\).
3. Dimension
- Dimension
- The dimension of a finite-dimensional vector space \(V\) is the length of any basis of the vector space, denoted by \(\dim V\)
If \(U_{1}, U_{2}\) are subspaces of a finite-dimensional vector space, then \[ \dim (U_{1} + U_{2}) = \dim U_{1} + \dim U_{2} - \dim (U_{1} \cap U_{2}) \]
Suppose \(U_{1}, \dots, U_{m}\) are finite-dimensional subspaces of \(V\) such that \(U_{1}+U_{2}+\dots+U_{m}\) is a direct sum. Prove that \(U_{1} \oplus U_{2} \oplus \dots \oplus U_{m}\) is finite-dimensional and that
\[ \dim U_{1} \oplus U_{2} \oplus \dots \oplus U_{m} = \sum_{i=1}^{m} \dim U_{i} \]