Exercises – Short Day

Exercises – Short Day#

Exercise 1: Ordered Bases in \(\mathbb{C^2}\)#

Is

\[\begin{split}\beta=\left(\left[\begin{array}{r}1\\i\end{array}\right],\left[\begin{array}{r}1+i\\-1+i\end{array}\right]\right) \end{split}\]

an ordered basis for \(\mathbb{C}^2\)?

Hint

Are the two vectors in \(\beta\) linearly independent?

Answer

The two vectors in \(\beta\) are linearly dependent over \(\mathbb{C}\) since

\[\begin{split}(1+i)\left[\begin{array}{r}1\\i\end{array}\right]=\left[\begin{array}{r}1+i\\-1+i\end{array}\right].\end{split}\]

Hence, \(\beta\) cannot be an ordered basis.

Exercise 2: Coordinates with respect to the Ordered Standard Basis#

The ordered standard basis for \(\mathbb{R}^3\) is given as \(\epsilon=({\mathbf e}_1,{\mathbf e}_2,{\mathbf e}_3)\), where

\[\begin{split} {\mathbf e}_1=\left[\begin{array}{r}1\\0\\0\end{array}\right],\, {\mathbf e}_2=\left[\begin{array}{r}0\\1\\0\end{array}\right],\, {\mathbf e}_3=\left[\begin{array}{r}0\\0\\1\end{array}\right]\, .\end{split}\]

Why does it hold true that

\[\begin{split} \left[\begin{array}{r}a\\b\\c\end{array}\right]_\epsilon = \left[\begin{array}{r}a\\b\\c\end{array}\right] \end{split}\]

for all \(a,b,c \in \mathbb{R}\)?

Consider why it more generally applies that \([{\mathbf v}]_\epsilon={\mathbf v}\) for all \(\mathbf v \in \mathbb{F}^m\) if \(\epsilon\) is the ordered standard basis for \(\mathbb{F}^m\) as described in Example 9.2.1.

Hint

According to Definition 9.5.1, the three entries in the \(\epsilon\) coordinate vector of a vector \(\mathbf v \in \mathbb{R}^3\) constitute the solution to the inhomogeneous linear equation system

\[c_1\cdot{\mathbf e}_1+c_2\cdot{\mathbf e}_2+c_3\cdot{\mathbf e}_3={\mathbf v} \, .\]

Now write

\[\begin{split} {\mathbf v}=\left[\begin{array}{r}a\\b\\c\end{array}\right] \end{split}\]

and try solving for \(c_1,c_2,c_3\).

Answer

See Example 9.5.1 in the textbook for the general case mentioned in the last sentence above.

Exercise 3: Span, Ordered Basis, and Coordinates#

We consider \(W=\mathrm{Span}({\mathbf w}_1,{\mathbf w}_2,{\mathbf w}_3,{\mathbf w}_4)\) in \(\mathbb{C}^4\) where

\[\begin{split} {\mathbf w}_1=\left[\begin{array}{c}0\\0\\0\\0\end{array}\right],\, {\mathbf w}_2=\left[\begin{array}{c}0\\i\\0\\-2\end{array}\right],\, {\mathbf w}_3=\left[\begin{array}{c}0\\-i\\0\\2\end{array}\right],\, {\mathbf w}_4=\left[\begin{array}{c}1\\1\\-1\\2i\end{array}\right] \, \, .\end{split}\]

Question a#

Determine an ordered basis for \(W\).

Answer

By using Theorem 9.2.1 in the textbook we find that \(({\mathbf w}_2,{\mathbf w}_4)\) is an ordered basis for \(W\).

Question b#

We are given the vector:

\[\begin{split}{\mathbf w}=\left[\begin{array}{c}1\\i\\-1\\-2\end{array}\right] \, .\end{split}\]

Calculate \([{\mathbf w}]_\beta\), where \(\beta=({\mathbf w}_2,{\mathbf w}_4)\).

Hint

Which linear equation system in the unknowns \(c_1\) and \(c_2\) does Definition 9.5.1 in the textbook give rise to?

Answer

\[\begin{split} [{\mathbf w}]_\beta=\left[\begin{array}{c}1+i\\1\end{array}\right] \, . \end{split}\]

Question c#

We are being informed that the equation \(c_1 {\mathbf w}_2 + c_2 {\mathbf w}_4 = {\mathbf u}\) in the unknowns \(c_1\) and \(c_2\) for some vector \(\mathbf u \in \mathbb{C}^4\) has no solutions. Can we conclude that \({\mathbf u}\) is not in \(W\)?

Answer

Yes.

Exercise 4: Ordered Bases and Coordinates in \(\mathbb{R}^4\)#

In \(\mathbb{R}^4\) we are given the five vectors:

\[\begin{split} {\mathbf v}_1=\left[\begin{array}{r}1\\-1\\2\\1\end{array}\right],\, {\mathbf v}_2=\left[\begin{array}{r}0\\1\\1\\3\end{array}\right],\, {\mathbf v}_3=\left[\begin{array}{r}1\\-2\\2\\-1\end{array}\right],\, {\mathbf v}_4=\left[\begin{array}{r}0\\1\\-1\\3\end{array}\right], \, {\mathbf v}=\left[\begin{array}{r}1\\-2\\2\\-3\end{array}\right] \, .\end{split}\]

We are being informed that

\[\begin{split}\mathrm{det}\left( \left[\begin{array}{rrrr}1 & 0 & 1 & 0\\-1 & 1 & -2 & 1\\2 & 1 & 2 & -1\\1 & 3 & -1 & 3\end{array}\right] \right) = 2.\end{split}\]

Question a#

Justify that \(\beta=({\mathbf v}_1,{\mathbf v}_2,{\mathbf v}_3,{\mathbf v}_4)\) is an ordered basis for \(\mathbb{R}^4\,\).

Hint

Can the given determinant be of use in some way?

Question b#

Determine the \(\beta\) coordinate vector \([{\mathbf v}]_\beta\), where \(\beta=({\mathbf v}_1,{\mathbf v}_2,{\mathbf v}_3,{\mathbf v}_4)\) is the ordered basis from Question a.

Hint

According to Definition 9.5.1, the four entries in the \(\beta\) coordinate vector for \(\mathbf v\) constitute the solution to the inhomogeneous linear equation system

\[c_1\cdot{\mathbf v}_1+c_2\cdot{\mathbf v}_2+c_3\cdot{\mathbf v}_3+c_4\cdot{\mathbf v}_4={\mathbf v} \, .\]

Therefore, try solving this equation system.

Answer

\[\begin{split} [{\mathbf v}]_\beta=\left[\begin{array}{r}2\\-1\\-1\\-1\end{array}\right]\, . \end{split}\]

Exercise 5: Coordinates with respect to an Ordered Basis#

We are given the vector

\[\begin{split}{\mathbf u}=\left[\begin{array}{r}1\\2\\3\end{array}\right]\end{split}\]

in \(\mathbb C^3\). Find an ordered basis \(\gamma\) for \(\mathbb{C}^3\) such that

\[\begin{split}[{\mathbf u}]_\gamma=\left[\begin{array}{r}1\\1\\1\end{array}\right].\end{split}\]

Exercise 6: Examples of Spans#

We are given a natural number \(n\) and an integer \(d\) that fulfill \(0 \le d \le n\). Find \(d\) vectors in \(\mathbb{R}^n\) such that their span has a dimension equal to \(d\).

Contents