Su'aalo Tusaale ah oo Ka Hadlaya Xiriirka Ka Dhexeeya Matrices iyo Isbeddellada
Pendahuluan
Matrix waa tirooyin ama walxo laydi ah oo loo habeeyey saf iyo tiirar. Matrices-ku si weyn ayaa loogu isticmaalaa dhinacyo kala duwan sida tirakoobka, fiisigiska, dhaqaalaha, iyo gaar ahaan isbeddellada joomatari ee xisaabta iyo sawirada kombiyuutarka. Matrices-ku waxay sidoo kale bixiyaan qalab wax ku ool ah oo lagu maareeyo xogta iyo si loogu sharaxo loona xalliyo dhibaatooyinka xisaabta ee kala duwan. Mid ka mid ah codsiyada muhiimka ah ee matrices-ku waa isbeddellada toosan, halkaas oo hawlgallada matrix-ka loo isticmaalo in lagu beddelo qaabka iyo booska walxaha joomatari ee booska.
Maqaalkan, waxaan ka hadli doonnaa tusaalooyin dhibaatooyin ah oo muujinaya sida loo isticmaalo matrices-ka isbeddellada toosan, waxaanan si faahfaahsan u sharxi doonnaa xalalkooda.
Qeexitaannada iyo Qoraallada
Si aan u bilowno, aan dib u eegno qaar ka mid ah qeexitaannada aasaasiga ah iyo qoraallada loo isticmaali doono dooddan:
1. Matrix: Tiro afargeesle ah oo loo habeeyey saf iyo tiirar.
2. Isbeddelka Toosan: Shaqada oo qaada vektor oo u maabeysa vektor kale iyadoo la adeegsanayo hawlgallada matrix.
3. Vektor: Waa walax ka mid ah set-ka vektor-ka oo leh dherer iyo jihayn, oo badanaa loo matalo tiir ama saf ku jira matrix.
Qorista Matrix-ka guud ahaan waxaa lagu qoraa xarfo waaweyn, tusaale ahaan \( A \), \( B \), vector-yadana waxaa lagu qoraa xarfo madow ama fallaadho korkooda ah, tusaale ahaan \( \mathbf{v} \) ama \( \vec{v} \).
Su'aalo iyo Doodo Tusaale ah
Su'aal 1aad: Isbeddelka Wareegga
Marka la eego shaxda isbeddelka wareegga \( R \) xagal \( \theta \) oo ku jirta booska laba-geesoodka ah:
\[ R = \begin{pmatrix} \cos\theta & -\sin\theta \\ sin\theta & \cos\theta \end{pmatrix} \]
Vektor \( \mathbf{v} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \). Go'aami natiijada isbeddelka vektor \( \mathbf{v} \) iyadoo la adeegsanayo matrix \( R \) haddii \( \theta = \frac{\pi}{2} \).
Dood:
Marka hore, geli qiimaha xagasha \( \theta = \frac{\pi}{2} \) shaxda \( R \):
\[ R = \begin{pmatrix} \cos\frac{\pi}{2} & -\sin\frac{\pi}{2} \\sin\frac{\pi}{2} & \cos\frac{\pi}{2} \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \]
Marka xigta, ku dhufo matrix \( R \) vector \( \mathbf{v} \):
\[ R \mathbf{v} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} (0 \cdot 1) + (-1 \cdot 0) \\ (1 \cdot 1) + (0 \cdot 0) \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \]
Markaa, natiijada ka dhalatay beddelka vektorka \( \mathbf{v} \) iyadoo la adeegsanayo matrixka \( R \) ee xagasha \( \theta = \frac{\pi}{2} \) waa vektorka \( \mathbf{v'} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \).
Su'aal 2aad: Isbeddelka Miisaanka
Marka la eego shaxda isbeddelka miisaanka \( S \) ee booska laba-geesoodka ah sida soo socota:
\[ S = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} \]
Vektor \( \mathbf{u} = \begin{pmatrix} 1 \\ 2 \end{pmatrix} \). Soo hel natiijada isbeddelka vektor \( \mathbf{u} \) iyadoo la adeegsanayo matrix \( S \).
Dood:
Ku dhufo matrix \( S \) vector \( \mathbf{u} \):
\[ S \mathbf{u} = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} (2 \cdot 1) + (0 \cdot 2) \\ (0 \cdot 1) + (3 \cdot 2) \end{pmatrix} = \begin{pmatrix} 2 \\ 6 \end{pmatrix} \]
Markaa, natiijada ka dhalatay beddelka vektorka \( \mathbf{u} \) iyadoo la adeegsanayo matrixka \( S \) waa vektorka \( \mathbf{u'} = \begin{pmatrix} 2 \\ 6 \end{pmatrix} \).
Su'aal 3: Isbeddelka Milicsiga
Marka la eego shaxda milicsiga \( F \) marka loo eego dhidibka y:
\[ F = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \]
Xisaabi natiijada beddelka vektorka \( \mathbf{w} = \begin{pmatrix} 3 \\ 4 \end{pmatrix} \) adoo isticmaalaya matrix-ka milicsiga \( F \).
Dood:
Ku dhufo matrix \( F \) vector \( \mathbf{w} \):
\[ F \mathbf{w} = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 3 \\ 4 \end{pmatrix} = \begin{pmatrix} (-1 \cdot 3) + (0 \cdot 4) \\ (0 \cdot 3) + (1 \cdot 4) \end{pmatrix} = \begin{pmatrix} -3 \\ 4 \end{pmatrix} \]
Markaa, natiijada ka dhalatay beddelka vektorka \( \mathbf{w} \) iyadoo la adeegsanayo matrixka \( F \) waa vektorka \( \mathbf{w'} = \begin{pmatrix} -3 \\ 4 \end{pmatrix} \).
Su'aal 4: Isbeddello Isku-dhafan
Ka soo qaad in ay jiraan laba matrices oo isbeddel ah, matrix wareeg ah oo ah xagal \( \theta = \frac{\pi}{4} \) iyo matrix miisaan ah oo ah S \) sida soo socota:
\[ R = \begin{pmatrix} \cos\frac{\pi}{4} & -\sin\frac{\pi}{4} \\sin\frac{\pi}{4} & \cos\frac{\pi}{4} \end{pmatrix} = \begin{pmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{pmatrix} \]
\[ S = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} \]
Isku dar isbeddelladan oo ku dabaq vektorka \( \mathbf{z} = \begin{pmatrix} 1 \\ 1 \end{pmatrix} \).
Dood:
Marka hore, xisaabi shaxda isbeddelka ee isku dhafan \( RS \):
\[ RS = R \cdot S = \begin{pmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{pmatrix} \cdot \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} = \begin{pmatrix} (\frac{\sqrt{2}}{2} \cdot 2) + (-\frac{\sqrt{2}}{2} \cdot 0) & (\frac{\sqrt{2}}{2} \cdot 0) + (-\frac{\sqrt{2}}{2} \cdot 0) + (-\frac{\sqrt{2}}{2} \cdot 3) \\ (\frac{\sqrt{2}}{2} \cdot 2) + (\frac{\sqrt{2}}{2} \cdot 0) & (\frac{\sqrt{2}}{2} \cdot 0) + (\frac{\sqrt{2}}{2} \cdot 3) \end{pmatrix} = \begin{pmatrix} \sqrt{2} & -\frac{3\sqrt{2}}{2} \\ \sqrt{2} & \frac{3\sqrt{2}}{2} \end{pmatrix} \]
Kadib, ku dhufo matrix-ka isku-darka ah \( RS \) vector-ka \( \mathbf{z} \):
\[ RS \mathbf{z} = \begin{pmatrix} \sqrt{2} & -\frac{3\sqrt{2}}{2} \\ \sqrt{2} & \frac{3\sqrt{2}}{2} \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} (\sqrt{2} \cdot 1) + (-\frac{3\sqrt{2}}{2} \cdot 1) \\ (\sqrt{2} \cdot 1) + (\frac{3\sqrt{2}}{2} \cdot 1) \end{pmatrix} = \begin{pmatrix} \sqrt{2} – \frac{3\sqrt{2}}{2} \\sqrt{2} + \frac{3\sqrt{2}}{2} \end{pmatrix} \]
Markaa, natiijada isbeddelka isku-dhafka ah ee vektorka \( \mathbf{z} \) iyadoo loo marayo matrixka \( RS \) waa:
\[ \mathbf{z'} = \begin{pmatrix} \frac{2\sqrt{2} – 3\sqrt{2}}{2} \\ \sqrt{2} + \frac{3\sqrt{2}}{2} \end{pmatrix} = \begin{pmatrix} -\frac{\sqrt{2}}{2} \\ \frac{5\sqrt{2}}{2} \end{pmatrix} \]
Gabagabo
Maqaalkan, waxaan ka wada hadalnay dhowr tusaale oo dhibaatooyin ah oo muujinaya sida loo isticmaalo matrices-ka isbeddellada toosan. Isbeddellada matrix-ku waxay door muhiim ah ka ciyaaraan dhinacyo badan, gaar ahaan sawirada kombiyuutarka iyo falanqaynta xogta. Annagoo fahmayna aasaaska isbeddellada matrix-ka, sida wareegga, cabbirka, iyo milicsiga, waxaan u gudbi karnaa inaan ku dabaqno fikradahan dhibaatooyin aad u adag. Barashada fikradahan waxay noqon doontaa mid qiimo leh qof kasta oo ka shaqeeya xisaabta, fiisigiska, ama sayniska kombiyuutarka.
