MATH 323 Lecture 12

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Substitute lecture

Chapter 3.4: Basis and Dimension

Definitions

We say that vectors ${\vec {v}}_{1},\ldots ,{\vec {v}}_{k}$ are linearly independendent if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_1 \, \vec{v}_1 + \dots + a_k \, \vec{v}_k = \vec{0}}

We say that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{w}_1, \ldots \vec{w}_k \in V} span Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} if any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{u} \in V} can be written as a linear combination of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {w}}_{1},\ldots ,{\vec {w}}_{k}} .

Let $V$ be a vector space.

The set ${\vec {e}}_{1},\ldots ,{\vec {e}}_{n}\in V$ is called a basis in $V$ if

1. ${\vec {e}}_{1},\ldots ,{\vec {e}}_{n}$ are linearly independent
2. They span $V$

The number of vectors in the basis is called the dimension of $V$

Theorem

All bases in $V$ have the same number of elements called the dimension of $V$.

Note: it is possible for a "finitely defined" Vector space to have infinite dimension.

Theorem 3.4.1

If $\{v_{1},\ldots ,v_{n}\}$ is a spanning set of $V$, $m>n$, then any vectors $w_{1},\ldots ,w_{m}$ are linearly independent.

Proof. (substitution/elimination process).

If $w_{j}$, $1\leq j\leq m$ is a 0 vector, then there is nothing to prove:

$0{\vec {w}}_{1}+\dots +\alpha {\vec {w}}_{j}+\dots +0{\vec {w}}_{m}={\vec {0}}$

If ${\vec {w}}_{1}\neq {\vec {0}}$, then ${\vec {w}}_{1}=a_{1}\,{\vec {v}}_{1}+\dots +a_{n}\,{\vec {v}}_{n}$. At least one of the $a$'s is nonzero. Without loss of generality, we assume that it is $a_{1}$ which is nonzero.

Therefore ${\vec {v}}_{1}={\frac {{\vec {w}}_{1}-a_{2}{\vec {v}}_{2}-\dots -a_{n}\,{\vec {v}}_{n}}{a_{1}}}$, so $\{{\vec {w}}_{1},{\vec {v}}_{2},\ldots ,{\vec {v}}_{n}\}$ is a spanning set for $V$.

Repeat the process with the above spanning set and ${\vec {w}}_{2},\ldots ,{\vec {w}}_{n}$. When we can no longer continue, we end up with $\{{\vec {w}}_{1},\ldots ,{\vec {w}}_{n}\}$ as a spanning set for $V$, and $\{{\vec {w}}_{n+1}=k_{1}\,{\vec {w}}_{1}+\dots +k_{n}\,{\vec {w}}_{n}\}$

Example

$\mathbb {R} ^{2}$ has bases Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {e}}_{1}=\left\langle 1,0\right\rangle } , ${\vec {e}}_{2}=\left\langle 0,1\right\rangle$ and ${\vec {v}}_{1}=\left\langle 1,1\right\rangle$, ${\vec {v}}_{2}=\left\langle 1,0\right\rangle$.

Let ${\vec {w}}_{1}=\left\langle 1,2\right\rangle$, ${\vec {w}}_{2}=\left\langle 2,3\right\rangle$, and ${\vec {w}}_{3}=\left\langle 3,4\right\rangle$

Note: $\{{\vec {e}}_{1},{\vec {e}}_{2}\}$ is a spanning set of $\mathbb {R} ^{2}$.

$w_{1}=1{\vec {e}}_{1}+2{\vec {e}}_{2}\quad \rightarrow \quad {\vec {e}}_{1}={\vec {w}}_{1}-2{\vec {e}}$

…

Corollary 3.4.2

If $\{v_{1},\ldots ,v_{n}\}$ and $\{w_{1},\ldots ,w_{m}\}$ are two bases in $V$, then $m=n$.

Proof. Assume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} 's are spanning set and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m > n} , then from Thm. 3.4.1, we know that $w_{1},\ldots ,w_{m}$ are linearly dependent. Not a basis (contradiction)

Flip Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n > m} and by same logic → contradiction. Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m=n} .

Infinite Dimension

If the vector space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} has no basis of finitely many vectors, we say that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} has infinite dimension.

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V = \{\vec{0}\}} , we say that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim V = 0}

Example

Find the dimension of the space of solutions to the system

${\begin{aligned}x+y+z&=0\\x-2y+3z&=0\end{aligned}}$

Three variables, two constraints: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3-2 = 1}

Note that the solutions form a vector space, and the set of solutions will be the nullspace of the coefficient matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix}1&1&1\\1&-2&3\end{bmatrix}} .

The solution is of the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha \left\langle -5, 2, 3 \right\rangle} , so the basis has only one vector. Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim N = 1}

Example

Find the dimension of the nullspace for the operation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = \left( x \frac{\mathrm{d}}{\mathrm{d}x} - 1 \right)} over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_2}

In other words, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} does Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) \mapsto x \, \frac{\mathrm{d}p(x)}{\mathrm{d}x} - p(x)} . For a polynomial of degree ≤ 1, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a+bx \mapsto x (b) - (a+bx) = -a} . If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(a+bx) = 0} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = 0} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} is anything. Therefore, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N(A) = \{bx ~\mid~ b \in \mathbb{R} \}} , the basis is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{x\}} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim N(A) = 1} .

Example

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^{2\times2}}

A basis in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^{2\times2}} would be:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \vec{e}_1 &= \begin{bmatrix}1&0\\0&0\end{bmatrix} \vec{e}_2 &= \begin{bmatrix}0&1\\0&0\end{bmatrix} \vec{e}_3 &= \begin{bmatrix}0&0\\1&0\end{bmatrix} \vec{e}_4 &= \begin{bmatrix}0&0\\0&1\end{bmatrix} \end{align}}

Theorem 3.4.3

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} be a vector space, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim V = n > 0} . Then

1. any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} linearly independent vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_1, \ldots, f_n} span Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
2. any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} vectors that span Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} are linearly independent.

Proof.

1. assume they do not span Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} : there is a vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{g}} which is not a linear combination of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} 's, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{f_1, \ldots, f_n, g\}} would be linearly independent. this contradicts #Theorem 3.4.1 since there are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} vectors with only Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} in a basis.
2. assume they are linearly dependent: then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_1 \, f_1 + \dots + c_n \, f_n = \vec{0}} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_1, \ldots, c_n} are not all 0. This means that one vector could be written as a linear combination of other vectors, so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n-1} vectors would span Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} . This contradicts the definition of a basis, stating that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} vectors must be linearly independent.

Theorem 3.4.4

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim V = n > 0}

1. No set of fewer than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} vectors spans Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
2. Any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m < n} linearly independent vectors could be extended (by more vectors) to form a basis.
3. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_1, \ldots, v_N} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N > n} , span Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} , we can pare them down to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} vectors, which would be a basis.