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Fenster schliessen

What is the \(\text{span}\{V_1,...,V_n\}\)?

\(\{\alpha_1V_1,...,\alpha_nV_n:\alpha_1,...,\alpha_n\ \epsilon\ \mathbb{R}\}\)

Fenster schliessen

\(E_1=\begin{bmatrix}1&0\\0&0\end{bmatrix}, E_2=\begin{bmatrix}0&1\\0&0\end{bmatrix}, \\E_3=\begin{bmatrix}0&0\\1&0\end{bmatrix}, E_4=\begin{bmatrix}0&0\\0&1\end{bmatrix}\)

What is \(\text{span}\{E_1,E_2,E_3,E_4\}\)?

\(\)\(\{ \alpha_1\begin{bmatrix}1&0\\0&0\end{bmatrix}, \alpha_2\begin{bmatrix}0&1\\0&0\end{bmatrix}, \\\alpha_3\begin{bmatrix}0&0\\1&0\end{bmatrix}, \alpha_4\begin{bmatrix}0&0\\0&1\end{bmatrix}:\\ \alpha_1,\alpha_2,\alpha_3,\alpha_4\ \epsilon\ \mathbb{R} \}\\\quad\quad\quad\quad\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ||\\ \{ \begin{bmatrix}\alpha_1&0\\0&0\end{bmatrix}, \begin{bmatrix}0&\alpha_2\\0&0\end{bmatrix},\\ \begin{bmatrix}0&0\\\alpha_3&0\end{bmatrix}, \begin{bmatrix}0&0\\0&\alpha_4\end{bmatrix}: \\\alpha_1,\alpha_2,\alpha_3,\alpha_4\ \epsilon\ \mathbb{R} \}\)

Fenster schliessen

What is the criteria for the set

\(\{V_1,...,V_n\}\)

to be linearly independent?

Given \(\alpha_1V1+...+\alpha_nV_n=0\),

where \(\alpha_1,...,\alpha_n\ \epsilon\ \mathbb{R}\).

 

All \(\alpha_1=...=\alpha_n=0\)

and there is no \(\alpha_i\ne0\).

Fenster schliessen

\(​​\{(1,0,1),(1,0,1),(1,1,0)\}\)

Check for linear independence?

\(\begin{bmatrix}1&1&1\\0&0&1\\1&1&0\end{bmatrix},\ \ \text{det}(A)=\)

 \(1\begin{bmatrix}1&1\\0&0\end{bmatrix}- 1\begin{bmatrix}1&1\\1&1\end{bmatrix}=0\)

 

Since \(\text{det}(A)=0\)

the set is linearly dependent.

Fenster schliessen

\(\{(1,0,0),(1,0,1),(1,1,0)\}\)

Check for linear independence?

\(\begin{bmatrix}1&1&1\\0&0&1\\0&1&0\end{bmatrix},\ \ \text{det}(A)=\)

\(-1\begin{bmatrix}1&1\\0&1\end{bmatrix}=-1\)

 

Since \(\text{det}(A)\ne0\)

the set is linearly independent.

Fenster schliessen

\(\\A=\{(x,y,z)\epsilon\mathbb{R}^3:x+2z=0\}\)

Find \(\text{Dim}(A)\).

\(\\x=-2z\rightarrow (-2\alpha,\beta,\alpha)\\ \text{span}((-2,0,1),(0,1,0))\)

Thus

\(\text{Dim}(A)=2\)

Fenster schliessen

With a basis:

\(\{v_1,v_2,...,v_n\}\)

What is the coordinates of:

\(\vec v=\{\alpha_1v_1,...,\alpha_nv_n\}\)?

The coordinates are:

\(\{\alpha_1,\alpha_2,...,\alpha_n\}\)

Fenster schliessen

\(E=\{(1,3),(4,5)\}\)

Find \(\vec a\)if the coordinates of E is \((3,4)\).

\(\\3(1,3)+4(4,5)\\ (3+16,9+20) \\\ \\\vec a=(19,29)\)