# Lernkarten

Karten 20 Karten 1 Lernende English Mittelschule 11.10.2019 / 13.11.2019 Keine Angabe
0 Exakte Antworten 20 Text Antworten 0 Multiple Choice Antworten

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

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

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

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

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

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 $$\alpha_i\ne0$$.

$$​​\{(1,0,1),(1,0,1),(1,1,0)\}$$

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.

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

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.

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

Find $$\text{Dim}(A)$$.

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

Thus

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

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:

$$\{\alpha_1,\alpha_2,...,\alpha_n\}$$

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

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

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