Lineare Algebra Beispiele

$[\begin{array}{c} \cos (45) & 0 & \sin (45) \\ 0 & 1 & 0 \\ - \sin (45) & 0 & \cos (45) \end{array}] \cdot [\begin{array}{c} \sqrt{2} & 0 & 0 \\ 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{2} \end{array}]$

Schritt 1

Der genau Wert von $\cos (45)$ ist $\frac{\sqrt{2}}{2}$ .

$[\begin{array}{c} \frac{\sqrt{2}}{2} & 0 & \sin (45) \\ 0 & 1 & 0 \\ - \sin (45) & 0 & \cos (45) \end{array}] \cdot [\begin{array}{c} \sqrt{2} & 0 & 0 \\ 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{2} \end{array}]$

Schritt 2

Der genau Wert von $\sin (45)$ ist $\frac{\sqrt{2}}{2}$ .

$[\begin{array}{c} \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ 0 & 1 & 0 \\ - \sin (45) & 0 & \cos (45) \end{array}] \cdot [\begin{array}{c} \sqrt{2} & 0 & 0 \\ 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{2} \end{array}]$

Schritt 3

Der genau Wert von $\sin (45)$ ist $\frac{\sqrt{2}}{2}$ .

$[\begin{array}{c} \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ 0 & 1 & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \cos (45) \end{array}] \cdot [\begin{array}{c} \sqrt{2} & 0 & 0 \\ 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{2} \end{array}]$

Schritt 4

Der genau Wert von $\cos (45)$ ist $\frac{\sqrt{2}}{2}$ .

$[\begin{array}{c} \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ 0 & 1 & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \end{array}] \cdot [\begin{array}{c} \sqrt{2} & 0 & 0 \\ 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{2} \end{array}]$

Schritt 5

Multipliziere $[\begin{array}{c} \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \\ 0 & 1 & 0 \\ - \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \end{array}] [\begin{array}{c} \sqrt{2} & 0 & 0 \\ 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{2} \end{array}]$ .

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Schritt 5.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is $3 \times 3$ and the second matrix is $3 \times 3$ .

Schritt 5.2

Multipliziere jede Zeile in der ersten Matrix mit jeder Spalte in der zweiten Matrix.

$[\begin{array}{c} \frac{\sqrt{2}}{2} \sqrt{2} + 0 \cdot 0 + \frac{\sqrt{2}}{2} \cdot 0 & \frac{\sqrt{2}}{2} \cdot 0 + 0 \sqrt{2} + \frac{\sqrt{2}}{2} \cdot 0 & \frac{\sqrt{2}}{2} \cdot 0 + 0 \cdot 0 + \frac{\sqrt{2}}{2} \sqrt{2} \\ 0 \sqrt{2} + 1 \cdot 0 + 0 \cdot 0 & 0 \cdot 0 + 1 \sqrt{2} + 0 \cdot 0 & 0 \cdot 0 + 1 \cdot 0 + 0 \sqrt{2} \\ - \frac{\sqrt{2}}{2} \sqrt{2} + 0 \cdot 0 + \frac{\sqrt{2}}{2} \cdot 0 & - \frac{\sqrt{2}}{2} \cdot 0 + 0 \sqrt{2} + \frac{\sqrt{2}}{2} \cdot 0 & - \frac{\sqrt{2}}{2} \cdot 0 + 0 \cdot 0 + \frac{\sqrt{2}}{2} \sqrt{2} \end{array}]$

Schritt 5.3

Vereinfache jedes Element der Matrix durch Ausmultiplizieren aller Ausdrücke.

$[\begin{array}{c} 1 & 0 & 1 \\ 0 & \sqrt{2} & 0 \\ - 1 & 0 & 1 \end{array}]$