Mathematics II · Unit 6 · 4 hrs
Vector Space Continued
Exam-focused notes for Vector Space Continued (Mathematics II, MTH168): what the TU syllabus asks and how it has actually been tested, with 4 solved past questions from this unit.
What this unit covers
- Dimension of vector space and Rank
- Change of basis
- Applications to difference equations
- Applications to Markov Chains
Applications to difference equations
Show that the solutions of $y_{k+2} - 4y_{k+1} + 3y_k = 0$ are linearly independent. [5]
Homogeneous linear difference equation: $$y{k+2} - 4y{k+1} + 3yk = 0$$ Task: show the two fundamental solutions are linearly independent. Assume a solution of the form $yk = m^k$. Substituting: $$m^{k+2} - 4m^{k+1} + 3m^k = 0$$ Dividing by $m^k$ (with $m \n...
Full solved answer →Verify that $1k$, $-2^k$, $3^k$ are linearly independent signals. [5]
Three discrete-time signals: - $f1(k) = 1^k$ - $f2(k) = (-2)^k$ - $f3(k) = 3^k$ Task: verify they are linearly independent. Signals are linearly independent if $$C1 f1(k) + C2 f2(k) + C3 f3(k) = 0 \quad \text{for all } k$$ forces $C1 = C2 = C3 = 0$. $$C1(1)...
Full solved answer →Verify that $1k$, $(-2^k)$, $3k$ are linearly independent signals. [5]
Three signals to test for linear independence: $$f1(k) = 1^k, \quad f2(k) = (-2)^k, \quad f3(k) = 3^k$$ No other numeric data required. --- Signals $\{f1, f2, f3\}$ are linearly independent if $$C1 f1(k) + C2 f2(k) + C3 f3(k) = 0 \quad \text{for all } k$$ i...
Full solved answer →Change of basis
Let $B={b_1, b_2}$ and $C={c_1, c_2}$ be bases for a vector space $V$, and suppose $b_1 = -c_1 + 4c_2$ and $b_2 = 5c_1 - 3c_2$. Find the change of coordinate matrix for the vector space and find $[x]_C$ for $x = 5b_1 + 3b_2$. [5]
- Bases $B=\{b1,b2\}$, $C=\{c1,c2\}$ for vector space $V$ - $b1=-c1+4c2$ - $b2=5c1-3c2$ - $x=5b1+3b2$ --- $$[b1]C=\begin{bmatrix}-1\\4\end{bmatrix},\qquad [b2]C=\begin{bmatrix}5\\-3\end{bmatrix}$$ $$P{C\leftarrow B}=\big[\,[b1]C\ \ [b2]C\,\big]=\begin{bmatr...
Full solved answer →Make Unit 6 stick
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