[Finite-Dimensional Linear Algebra] 2.5 + 2.6 + 2.7 Basis and Dimension

Lemma For a set of vectors $u_1, u_2,\dots , u_n$ in vector space $V$, if $v \in sp \{ u_1, u_2,\dots u_n \}$, then

$$sp \{ u_1, u_2,\dots u_n, v \} = sp \{ u_1, u_2,\dots u_n \}$$

This leads to the concept of a basis, a spanning set containing the fewest possible elements. 

Definition of a Basis

Let $u_1, u_2,\dots u_n$ be vectors in a vector space $V$. We say that $\{ u_1, u_2,\dots u_n \}$ is a basis for $V$ iff  $\{ u_1, u_2,\dots , u_n \}$ is a spanning set for $V$ and is linearly independent.

Definition of Linear Independence

Let $V$ be a vector space over a field $F$. Let $u_1, u_2,\dots u_n$ be vectors in $V$, and let $\alpha _1, \alpha _2,\dots \alpha _n$  be scalars in $F$. We say that $\{ u_1, u_2,\dots , u_n \}$ is linearly independent iff   $$\alpha _1 u_1+\alpha _2u_2+\cdots + \alpha _nu_n = 0$$ only when $$\alpha _1 = \alpha _2 = \dots = \alpha _n =0.$$

• inversely, a set is linearly dependent iff there exists a nontrivial solution to $\alpha _1 = \alpha _2 = \dots = \alpha _n =0$ in which not all of the scalars $\alpha _i$ are zero. 

Figure 2.5.1. Visualization of linear independence

Intuitively, linearly independent vectors represent different directions in a vector space, such as the x-axis and the y-axis that only have one common element, 0. 

 

Theory A set of vectors $\{ u_1, u_2,\dots u_n \}$, where $n \geq 2,$ is linearly dependent iff at least one of the vectors can be written as a linear combination of the remaining $n-1$ vectors. 

 

Lemma If a set of vectors $\{ u_1, u_2,\dots u_n \}$ is linearly dependent, then there exists an integer $k$, with $2 \leq k \leq n$, such that $u_k$ is a linear combination of $u_1, u_2,\dots u_{k-1}$. 

 

Theory A set of vectors $\{ u_1, u_2,\dots u_n \}$ is linearly independent iff each vector in $sp \{ u_1, u_2,\dots u_n \}$ can be written uniquely as a linear combination of $u_1, u_2,\dots u_n$. 

 

Theory A set of vectors $\{ u_1, u_2,\dots u_n \}$ is a basis for $V$ iff each vector in $v \in V$ can be written uniquely as a linear combination of $u_1, u_2,\dots u_n$. 

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Dimensions of a Vector Space

Definition of Finite-Dimensionality

Let $V$ be a vector space over a field $F$. We say that $V$ is finite-dimensional if

• $V$ has a basis or
• $V$ is a trivial vector space $\{ 0 \}$.

If $V$ is not finite-dimensional, then it is called infinite-dimensional. 

 

Corollary Every nontrivial subspace of a finite-dimensional vector space has a basis

 

Theorem Let $V$ be a nontrivial vector space over a field $F$, and suppose $\{ u_1, u_2,\dots , u_m \}$ spans $V$. Then a subset of the spanning set is a basis for $V$. 

 

Theorem Let $V$ be a finite-dimensional vector space over a field $F$, and suppose $\{ u_1, u_2,\dots , u_k \}$ is linearly independent. If the linearly independent set of vectors does not span $V$, then there exist vectors $u_{k+1}, u_{k+2},\dots , u_n$ such that $$\{ u_1, u_2,\dots , u_k, u_{k+1}, \dots , u_n \}$$ is a basis for $V$. 

 

Theorem Let $V$ be a finite-dimensional vector space over a field $F$, and let $\{ u_1, u_2,\dots u_m \}$ be a basis for $V$. If $v_1, v_2,\dots v_n$ are any $n$ vectors in $V$, with $n > m$, then the set $\{ v_1, v_2,\dots v_n \}$ is linearly dependent. 

 

Corollary Let $\{ u_1, u_2,\dots u_m \}$ and $\{ v_1, v_2,\dots v_m \}$ be two bases for $V$. Then $n = m$.

Definition of Dimensions of a Vector Space

Let $V$ be a finite-dimensional vector space.

• If $V$ is a trivial vector space, then we say that the dimension of $V$ is zero. 
• Else, the dimension of $V$ is the number of vectors in a basis for $V$. 

 

Example The dimensions of $P_n$ is $n+1$ since $\{ 1, x, x^2, \dots x^n \}$ is a basis. 

 

Theorem Let V be an n-dimensional vector space over a field $F$, and let $u_1, u_2,\dots u_n$ be vectors in V. 

1. If $\{ u_1, u_2,\dots u_n \}$ spans $V$, then it is linearly independent and hence a basis for $V$.
2. If $\{ u_1, u_2,\dots u_n \}$ is linearly independent, then it spans $V$ and hence a basis for $V$. 

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basis: a minimal spanning set for a vector space