🔬 Tutorial problems delta \delta

🔬 Tutorial problems delta \(\delta\)#

Note

This problems are designed to help you practice the concepts covered in the lectures. Not all problems may be covered in the tutorial, those left out are for additional practice on your own.

\(\delta\).1#

This example appears to be part of an application of a linear version of a Keynesian cross model.

Solve the following system of linear equations:

\[\begin{split} \left\{\begin{array}{ccccc} 2 Y-5 C+0.8 T & = & 580 & \cdots & (\text { Equation } 1) \\ -Y+C+0.6 T+340 & = & 0 & \cdots & (\text { Equation } 2) \\ 0.4 Y-T & = & 100 & \cdots & (\text { Equation } 3) \end{array}\right\} \end{split}\]

[Bradley, 2013] Progress Exercises 9.3, Question 5.

It would be helpful to review Gauss-Jordan elimination technique, for example here

\(\delta\).2#

Find the inverse matrix for the following matrix or show that it does not exist:

\[\begin{split} C=\left(\begin{array}{ccc} 1 & 1 & -3 \\ 2 & 1 & -3 \\ 2 & 2 & 1 \end{array}\right) \end{split}\]

[Sydsæter, Hammond, Strøm, and Carvajal, 2016] Section 16.6, Problem 2

It would be helpful to review Gauss-Jordan elimination technique for computing inverse matrices, for example here

\(\delta\).3#

Consider the matrix \(A\) defined by

\[\begin{split} % A = \begin{pmatrix} 1 & 0 \\ 0.5 & -2 \\ 0 & 3 \end{pmatrix} % \end{split}\]

Do the columns of this matrix form a basis of \(\mathbb{R}^3\)?

Why or why not?

Check all relevant definitions and facts, and apply them

\(\delta\).4#

Is \(\mathbb{R}^2\) a linear subspace of \(\mathbb{R}^3\)?

Why or why not?

Check all relevant definitions and facts, and apply them

\(\delta\).5#

Show that if \(T \colon \mathbb{R}^K \to \mathbb{R}^N\) is a linear function then \(0 \in \mathrm{kernel}(T)\).

Check all relevant definitions and facts, and apply them

\(\delta\).6#

Let \(S\) be any nonempty subset of \(\mathbb{R}^N\) with the following two properties:

  • \(x, y \in S \implies x + y \in S\)

  • \(c \in \mathbb{R}\) and \(x \in S \implies cx \in S\)

Is \(S\) a linear subspace of \(\mathbb{R}^N\)?

Check all relevant definitions and facts, and apply them

\(\delta\).7#

If \(S\) is a linear subspace of \(\mathbb{R}^N\) then any linear combination of \(K\) elements of \(S\) is also in \(S\). Show this for the case \(K = 3\).

Check all relevant definitions and facts, and apply them

\(\delta\).8#

Let \(\{x_1, x_2\}\) be a linearly independent set in \(\mathbb{R}^2\) and let \(\gamma\) be a nonzero scalar.

Is it true that \(\{\gamma x_1, \gamma x_2\}\) is also linearly independent?

Check all relevant definitions and facts, and apply them

\(\delta\).9#

Is

\[\begin{split} z= \begin{pmatrix} -3.98 \\ 11.73 \\ -4.32 \end{pmatrix} \end{split}\]

in the span of \(X:=\{x_1, x_2, x_3\}\), where

\[\begin{split} x_1= \begin{pmatrix} -4 \\ 0 \\ 0 \end{pmatrix}, \;\; x_2= \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}, \;\; x_3= \begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}? \end{split}\]

Check all relevant definitions and facts, and apply them

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