diff --git a/matrixmystics.md b/matrixmystics.md
index 13c343f..6efe6bf 100644
--- a/matrixmystics.md
+++ b/matrixmystics.md
@@ -7,26 +7,61 @@ permalink: /matrixmystics/
* TOC
{:toc}
-# Module 1
-
+# Module A
___
-1. Ram and Lakshman were two brothers, Ram's pocket money was twice as much as Lakshman. The good boy that Ram and Lakshman were, they did not spend their pocket money on anything. They instead saved the same in their piggy bank. Every week, they would check their savings so far. Assume the first week's savings was $$(R_1,L_1)$$ and second week's $$(R_2,L_2)$$ and so on. They try plotting their weekly savings on a graph sheet. How will the points look like?
-2. Atul's house is centered at origin $$(0,0)$$ he walks straight (along the x-axis) for 2 units and then takes a left and walks 1 unit to reach Bala's house, after that he takes a right turn and walks for one unit and then a left turn and walks for one unit and reaches Chetan's house. He continues in a similar style, takes a right turn 1 unit and then left turn one unit and reaches Divya's house. Are the houses of Bala, Chetan and Divya on a straight line? What is the equation of this line? Plot this on Geogebra
+1.Ram and Lakshman were two brothers, Ram's pocket money was twice as much as Lakshman. The good boy that Ram and Lakshman were, they did not spend their pocket money on anything. They instead saved the same in their piggy bank. Every week, they would check their savings so far. Assume the first week's savings was $$(R_1,L_1)$$ and second week's $$(R_2,L_2)$$ and so on. They try plotting their weekly savings on a graph sheet. How will the points look like?
+
+ Sol.
+
+ Put your solution here.
+
+
+
+2.Atul's house is centered at origin $$(0,0)$$ he walks straight (along the x-axis) for 2 units and then takes a left and walks 1 unit to reach Bala's house, after that he takes a right turn and walks for one unit and then a left turn and walks for one unit and reaches Chetan's house. He continues in a similar style, takes a right turn 1 unit and then left turn one unit and reaches Divya's house. Are the houses of Bala, Chetan and Divya on a straight line? What is the equation of this line? Plot this on Geogebra
+
+ Sol.
+ Put your solution here.
+
-3. Plot the lines $$y=x$$, $$y=2x$$, $$y=10x$$.
-4. Observe that they all pass through the origin. Why?
+3.Plot the lines $$y=x$$, $$y=2x$$, $$y=10x$$.
+
+ Sol.
+
+ Put your solution here.
+
+
-5. Plot $$y=2x+1$$. Observe, Why doesn't it pass through the origin?
+4.Observe that they all pass through the origin. Why?
+
+ Sol.
+
+ Put your solution here.
+
+
-6. Plot $$y=ax+b$$, with $$a$$ and $$b$$ as parameters which you should be able to vary. What do you observe?
+5.Plot $$y=2x+1$$. Observe, Why doesn't it pass through the origin?
+
+ Sol.
+
+ Put your solution here.
+
+
-7. Consider the following simultaneous equation:
+6.Plot $$y=ax+b$$, with $$a$$ and $$b$$ as parameters which you should be able to vary. What do you observe?
+
+ Sol.
+
+ Put your solution here.
+
+
+
+7.Consider the following simultaneous equation:
$$2x+3y=7$$
$$3x+4y=10$$
@@ -44,47 +79,130 @@ ___
10\\
\end{matrix}
\right)$$
-8. Consider a simple function $$f(x) = 3x+2$$. This function is invertible right? Can you tell us what is $$\alpha$$ such that $$f(\alpha)=17$$? Is such an $$\alpha$$ unique? How did you find such an $$\alpha?$$. Is this always possible?
+
+ Sol.
+
+ Put your solution here.
+
+
+
+8.Consider a simple function $$f(x) = 3x+2$$. This function is invertible right? Can you tell us what is $$\alpha$$ such that $$f(\alpha)=17$$? Is such an $$\alpha$$ unique? How did you find such an $$\alpha?$$. Is this always possible?
+
+ Sol.
+
+ Put your solution here.
+
+
+
+9.Consider the function $$f(x)=x^2-10$$, what is $$f(5)$$?
+
+ Sol.
+
+ Put your solution here.
+
+
+
+10.Consider the function $$f(x)=x^2-10$$, if $$f(\alpha)=54$$, what is $$\alpha$$?
+
+ Sol.
+ Put your Solution here.
+
+
+11.Consider the function $$g(x)=x^3-x^2-10x+2$$, if $$g(x)=-22$$ what is $$x$$?
+
+ Sol.
+
+ Put your solution here.
+
+
+
+
+12.Do you know what is $$\mathbb{R}, \mathbb{R}^2 and\ \mathbb{R}^3$$ ?
+
+
+ Sol.
+
+ Put your solution here.
+
+
+
+
+13.Consider the function $$\phi : \mathbb{R}^2\rightarrow \mathbb{R}^2$$ defined by $$\phi (x,y)=(2x+3y,3x+4y)$$. What has this to do with the previous question?
+
+
+ Sol.
-9. Consider the function $$f(x)=x^2-10$$, what is $$f(5)$$?
+ Put your solution here.
-10. Consider the function $$f(x)=x^2-10$$, if $$f(\alpha)=54$$, what is $$\alpha$$?
+
-11. Consider the function $$g(x)=x^3-x^2-10x+2$$, if $$g(x)=-22$$ what is $$x$$?
+14.Is the function $$\phi$$ invertible? In the question above on matrices, we see that it is of the form $$A\vec{x}=b$$. Note that we can invert the matrix, using the method that was taught to us in our high school to find out the value for the variables $$x$$ and $$y$$. This is one of the many applications of matrices.
+
+ Sol.
-12. Do you know what is $$\mathbb{R}, \mathbb{R}^2 and\ \mathbb{R}^3$$ ?
+ Put your solution here.
+
-13. Consider the function $$\phi : \mathbb{R}^2\rightarrow \mathbb{R}^2$$ defined by $$\phi (x,y)=(2x+3y,3x+4y)$$. What has this to do with the previous question?
+15.We will now see matrices as functions. Instead of $$\phi$$ we will write the matrix itself:
$$\left( \begin{matrix} 2 & 3 \\3 & 4 \\\end{matrix}\right) : \mathbb{R}^2 \rightarrow \mathbb{R}^2$$.
-14. Is the function $$\phi$$ invertible? In the question above on matrices, we see that it is of the form $$A\vec{x}=b$$. Note that we can invert the matrix, using the method that was taught to us in our high school to find out the value for the variables $$x$$ and $$y$$. This is one of the many applications of matrices.
+
+ Sol.
-15. We will now see matrices as functions. Instead of $$\phi$$ we will write the matrix itself:
$$\left( \begin{matrix} 2 & 3 \\3 & 4 \\\end{matrix}\right) : \mathbb{R}^2 \rightarrow \mathbb{R}^2$$.
+ Put your solution here.
-16. Consider the function $$\left( \begin{matrix} 1 & 2 \\2 & 4 \\\end{matrix}\right) : \mathbb{R}^2 \rightarrow \mathbb{R}^2$$. This matrix takes a few elements to the origin. What are those elements? Plot this using Geogebra.
+
+16.Consider the function $$\left( \begin{matrix} 1 & 2 \\2 & 4 \\\end{matrix}\right) : \mathbb{R}^2 \rightarrow \mathbb{R}^2$$. This matrix takes a few elements to the origin. What are those elements? Plot this using Geogebra.
+
+ Sol.
-# Module 2
+ Put your solution here.
+
+
+
+
+
+# Module B
---
-1. $$A$$ is assigned $$0$$, $$B:1$$, $$C:2$$, and so on up to $$Z:25$$
+17.$$A$$ is assigned $$0$$, $$B:1$$, $$C:2$$, and so on up to $$Z:25$$
Assume you denoted every letter with a number, as given in the table above.
You need to encrypt the word $$SUDARSHANA$$ which stands for the numbers: $$18, 20, 3, 0, 17, 18, 7, 0, 13,0$$.
You encrypt this using a matrix given by : $$ \left( \begin{matrix} 2 & 3 \\3 & 4 \\\end{matrix}\right) $$.
So $$SUDARSHANA$$ will end up becoming: $$96, 134, 6, 9, 88, 123, 14, 21, 26, 39$$.
Given these numbers, how will you decrypt the message and get back $$SUDARSHANA$$?
. This is a well known cryptographic protocol called the Hill Cipher. You can read more online.
-1. We encounter equations very often in our lives. Consider for example, the following situation at ___Baker's Cafe___. The manager has a very important estimate to make. Mostly, visitors at his cafe happen to be families and they are often comprised of Children and/or Adults. He observes that there are 3 adults and 1 child at a table and their bill turns out to be Rs.1200/-. There is yet another table with 2 children and 1 adult and their bill comes out to be Rs.1000/-. Can the manager estimate the consumption of a Child/Adult? This is popularly called the _Simultaneous Equations_ and we all remember from our school days, multiple ways in which these can be solved.
+
+ Sol.
+
+ Put your solution here.
+
+
+
+18.We encounter equations very often in our lives. Consider for example, the following situation at ___Baker's Cafe___. The manager has a very important estimate to make. Mostly, visitors at his cafe happen to be families and they are often comprised of Children and/or Adults. He observes that there are 3 adults and 1 child at a table and their bill turns out to be Rs.1200/-. There is yet another table with 2 children and 1 adult and their bill comes out to be Rs.1000/-. Can the manager estimate the consumption of a Child/Adult? This is popularly called the _Simultaneous Equations_ and we all remember from our school days, multiple ways in which these can be solved.
$$ 3A + 1C = 1200 $$
$$ 1A + 2C = 1000 $$
+
+ Sol.
-2. While we were taught the so called two variables and two unknowns, what if there were more equations than unknowns?
+ Put your solution here.
+
+
+
+19.While we were taught the so called two variables and two unknowns, what if there were more equations than unknowns?
$$ 3A + 1C = 1200 $$
$$ 1A + 2C = 1000 $$
$$ 1A + 1C = 900 $$
+
+ Sol.
+
+ Put your solution here.
-3. Note that the previous question can be modelled as a matrix:
+
+
+20.Note that the previous question can be modelled as a matrix:
$$ 3A + 1C = 1200 $$
$$ 1A + 2C = 1000 $$
@@ -109,125 +227,313 @@ $$
\end{matrix}
\right)
$$
+
+ Sol.
+
+ Put your solution here.
+
+
+21.One obvious way to solve this, is to guess the values :-). Can you get closer to the solution by guessing? Note that there is no solution to this question. You can just reduce the error. Do you see why?
+
+ Sol.
+ Put your solution here.
-4. One obvious way to solve this, is to guess the values :-). Can you get closer to the solution by guessing? Note that there is no solution to this question. You can just reduce the error. Do you see why?
+
-5. In the figure below:
+22.In the figure below:
If 1000 people were to start in one state, what will be the distribution of people eventually?
+
+ Sol.
-5. In the figure below:
+ Put your solution here.
+
+
+
+23.In the figure below:
If 1000 people were to start in one state, what will be the distribution of people eventually?
+
+ Sol.
+
+ Put your solution here.
-# Module 3
+
+
+# Module C
___
-1. Use Geogebra:
+24.Use Geogebra:
Draw the vector $$\begin{bmatrix}1 \\
1 \\
\end{bmatrix}$$. Find out all those vectors which are perpendicular to this vector.
+
+ Sol.
+
+ Put your solution here.
+
+
-2. Do you observe that we are asking for vectors $$\begin{bmatrix}x \\
+25.Do you observe that we are asking for vectors $$\begin{bmatrix}x \\
y \\
\end{bmatrix}$$ such that,
$$\begin{bmatrix}1 & 1 \\
\end{bmatrix} \begin{bmatrix} x\\ y \\ \end{bmatrix}=0 $$
+
+ Sol.
-3. Use Geogebra and solve the above question with $$\begin{bmatrix} 1 \\
+ Put your solution here.
+
+
+
+26.Use Geogebra and solve the above question with $$\begin{bmatrix} 1 \\
1 \\
\end{bmatrix}$$ replaced by $$\begin{bmatrix}a \\
b \\
\end{bmatrix}$$.Use $$(a,b)$$ as parameters and check what happens to $$(x,y)$$.
+
+ Sol.
+
+ Put your solution here.
-4. What is $$(x,y,z)$$ satisfying the following equation? (Use Geogebra)
+
+
+27.What is $$(x,y,z)$$ satisfying the following equation? (Use Geogebra)
$$\begin{bmatrix}1 & 2 & 3\\
\end{bmatrix} \begin{bmatrix} x\\ y \\ z\\ \end{bmatrix}=0 $$
+
+ Sol.
+
+ Put your solution here.
+
+
-5. Use Geogebra and plot all the points in the set below.
+28.Use Geogebra and plot all the points in the set below.
$$T= \{ \alpha(1,2,1) | \alpha \in \mathbb{R}\} $$
+
+ Sol.
-6. Use Geogebra and plot all the points in the set below.
+ Put your solution here.
+
+
+
+29.Use Geogebra and plot all the points in the set below.
$$S= \{ \beta(2,7,3) | \beta \in \mathbb{R}\} $$
+
+ Sol.
+
+ Put your solution here.
+
+
-7. Use Geogebra and plot all the points in the set below.
+30.Use Geogebra and plot all the points in the set below.
$$W= \{\alpha(1,2,1) + \beta(2,7,3) | \alpha,\beta \in \mathbb{R}\} $$
+
+ Sol.
-8. In the above set $$W$$ find out all the points $$(x,y,z)$$ satisfying the following: (Use Geogebra)
+ Put your solution here.
+
+
+
+31.In the above set $$W$$ find out all the points $$(x,y,z)$$ satisfying the following: (Use Geogebra)
$$\begin{bmatrix}w_1 & w_2 & w_3\\
\end{bmatrix} \begin{bmatrix} x\\ y \\ z\\ \end{bmatrix}=0 $$
where $$(w_1,w_2,w_3) \in W$$. Note that $$w_i$$s are real numbers.
+
+ Sol.
+
+ Put your solution here.
-9. Given the matrix $$A=\begin{bmatrix}1 & 2 & 3\\ 4 & 5 &6\\ 7 & 8 & 9\\ \end{bmatrix}$$, find out all the possible $$(x,y,z)$$ such that:
+
+
+32.Given the matrix $$A=\begin{bmatrix}1 & 2 & 3\\ 4 & 5 &6\\ 7 & 8 & 9\\ \end{bmatrix}$$, find out all the possible $$(x,y,z)$$ such that:
$$ \begin{bmatrix}1 & 2 & 3\\ 4 & 5 &6\\ 7 & 8 & 9\\ \end{bmatrix} \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix}=0$$
- Observe carefully, what has this question got to do with previous five questions in this module
+Observe carefully, what has this question got to do with previous five questions in this module
+
+ Sol.
+
+ Put your solution here.
+
+
-10. Given the matrix $$A=\begin{bmatrix}1 & 2 & 3\\ 4 & 5 &6\\ 7 & 8 & 9\\ \end{bmatrix}$$ what does the following three sets represent?
+33.Given the matrix $$A=\begin{bmatrix}1 & 2 & 3\\ 4 & 5 &6\\ 7 & 8 & 9\\ \end{bmatrix}$$ what does the following three sets represent?
(i) $$\mathscr{R}=\{\alpha(1,2,3) + \beta(4,5,6) + \gamma(7,8,9) |\alpha, \beta, \gamma\in \mathbb{R}\} $$
(ii) $$C=\{\alpha(1,4,7) + \beta(2,5,8) + \gamma(3,6,9) | \alpha, \beta, \gamma \in \mathbb{R}\}$$
(iii) $$N=\{(x,y,z)|x(1,4,7) + y(2,5,8) + z(3,6,9) = 0 \}$$
Use only Geogebra :)
+
+ Sol.
+
+ Put your solution here.
+
+
+
+34.Did you observe that every vector of $$\mathscr{R}$$ is perpendicular to every vector of $$N$$?
+
+ Sol.
-11. Did you observe that every vector of $$\mathscr{R}$$ is perpendicular to every vector of $$N$$?
+ Put your solution here.
-12. Consider the matrix $$B=\begin{bmatrix} 1 & 2 \\ 2 & 4\\ \end{bmatrix}$$. Draw the line $$2y+x=4$$. Seeing the matrix $$B$$ as a function $$B:\mathbb{R^2}\mapsto \mathbb{R^2}$$,
where does $$B$$ takes the line $$2y+x=4$$?
+
+
+35.Consider the matrix $$B=\begin{bmatrix} 1 & 2 \\ 2 & 4\\ \end{bmatrix}$$. Draw the line $$2y+x=4$$. Seeing the matrix $$B$$ as a function $$B:\mathbb{R^2}\mapsto \mathbb{R^2}$$,
where does $$B$$ takes the line $$2y+x=4$$?
Where does it take:
i)$$2y+x=10$$
ii)$$2y+x=62$$
iii)$$2y+x=1800$$
+
+ Sol.
+
+ Put your solution here.
+
+
+
+36.In general $$B=\begin{bmatrix} 1 & 2 \\ 2 & 4\\ \end{bmatrix}:\mathbb{R^2}\mapsto \mathbb{R^2}$$, where does this function take $$2y+x=k$$?(where $$k$$ is a constant)
+
+ Sol.
+
+ Put your solution here.
+
+
-13. In general $$B=\begin{bmatrix} 1 & 2 \\ 2 & 4\\ \end{bmatrix}:\mathbb{R^2}\mapsto \mathbb{R^2}$$, where does this function take $$2y+x=k$$?(where $$k$$ is a constant)
+37.Given $$B=\begin{bmatrix} 1 & 2 \\ 2 & 4\\ \end{bmatrix}:\mathbb{R^2}\mapsto \mathbb{R^2}$$. What is the range of this function?
+
+ Sol.
-14. Given $$B=\begin{bmatrix} 1 & 2 \\ 2 & 4\\ \end{bmatrix}:\mathbb{R^2}\mapsto \mathbb{R^2}$$. What is the range of this function?
+ Put your solution here.
-15. You have achieved the required wisdom if you have realized that:
+
+
+38.You have achieved the required wisdom if you have realized that:
$$B=\begin{bmatrix} 1 & 2 \\ 2 & 4\\ \end{bmatrix}:\mathbb{R^2}\mapsto \mathbb{R^2}$$.
" $$B$$ collapses a dimension ".
+
+ Sol.
+
+ Put your solution here.
+
+
-# Module 4
+# Module D
-1. Given the matrix $$M=\begin{bmatrix} 1 & 3 \\ 2 & 6\\ \end{bmatrix}$$. Use Geogebra to plot $$\mathscr{R}$$, $$\mathscr{C}$$ & $$\mathscr{N}$$. what do you observe?
+---
+
+39.Given the matrix $$M=\begin{bmatrix} 1 & 3 \\ 2 & 6\\ \end{bmatrix}$$. Use Geogebra to plot $$\mathscr{R}$$, $$\mathscr{C}$$ & $$\mathscr{N}$$. what do you observe?
(i) $$\mathscr{R}=\{\alpha(1,3) + \beta(2,6) | \alpha, \beta\in \mathbb{R}\} $$
(ii) $$\mathscr{C}=\{\alpha(1,2) + \beta(3,6) | \alpha, \beta\in \mathbb{R}\} $$
(iii) $$\mathscr{N}=\{(x,y)| x(1,3) + y(2,6) = 0, \forall x,y\in \mathbb{R} \}$$
+
+ Sol.
+
+ Put your solution here.
-2. Note that $$\mathscr{R}$$ and $$\mathscr{N}$$ are orthogonal.
+
-3. What is the null-space of $$M=\begin{bmatrix} 1 & 3 \\ 2 & 6\\ \end{bmatrix}$$ & the null-space of $$M^T$$?
+40.Note that $$\mathscr{R}$$ and $$\mathscr{N}$$ are orthogonal.
+
+ Sol.
-4. Do you observe that $$C(M)$$ ⊥ $$N(M^T)$$ , $$R(M)$$ ⊥ $$N(M)$$ ?
+ Put your solution here.
-5. Consider $$A=\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{bmatrix}$$. What is $$N(A)$$, $$C(A)$$, $$R(A)$$, $$N(A^T)$$.
+
-6. Consider a 4x4 matrix $$M$$: $$\mathbb{R^4}\mapsto \mathbb{R^4}$$ whose range is
+41.What is the null-space of $$M=\begin{bmatrix} 1 & 3 \\ 2 & 6\\ \end{bmatrix}$$ & the null-space of $$M^T$$?
+
+ Sol.
+
+ Put your solution here.
+
+
+
+42.Do you observe that $$C(M)$$ ⊥ $$N(M^T)$$ , $$R(M)$$ ⊥ $$N(M)$$ ?
+
+ Sol.
+
+ Put your solution here.
+
+
+
+43.Consider $$A=\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{bmatrix}$$. What is $$N(A)$$, $$C(A)$$, $$R(A)$$, $$N(A^T)$$.
+
+ Sol.
+
+ Put your solution here.
+
+
+
+44.Consider a 4x4 matrix $$M$$: $$\mathbb{R^4}\mapsto \mathbb{R^4}$$ whose range is
a) $$4-Dimension$$
b) $$3-Dimension$$
c) $$2-Dimension$$
d) $$1-Dimension$$
e) $$0-Dimension$$
Give an example each for all the above 5 cases.
+
+ Sol.
+
+ Put your solution here.
-7. Consider $$A : \mathbb{R^{3}} \to \mathbb{R^{3}} $$
+
+
+45.Consider $$A : \mathbb{R^{3}} \to \mathbb{R^{3}} $$
a) Show that if the range contains a point $$(a,b,c)$$, then it should contain the entire set $$S$$, defined by:
$$S= \{\alpha(a, b,c)| \alpha \in \mathbb{R}\}$$.
b) Show that if the range contains the points $$(a,b,c)$$ and $$(d,e,f)$$, then the range contains the entire set $$T$$ defined by:
$$T=\{\alpha(a,b,c) + \beta(d,e,f) \thinspace|\thinspace \alpha,\beta\in \mathbb{R}\}$$.
c) Note: $$S$$ is of the dimension $$1$$, but $$T$$ need'nt be of dimension $$2$$. Think!
+
+ Sol.
+
+ Put your solution here.
+
+
+
+# Module E
+
+---
+
+46.Give an example of two $$2-dim$$ subspaces in $$\mathbb{R^{3}}$$. Let us call it $$S_1, S_2$$.
+
+ Sol.
+
+ Put your solution here.
+
+
+
+
+47.Let $$S_3$$ be all those vectors perpendicular to $$S_1$$. $$S_4$$ be that of $$S_2$$.
+
+ Sol.
+
+ Put your solution here.
+
+
+
+
+48.Find a matrix $$M$$ whose Null-Space is $$S_3$$. column space is $$S_2$$.
+
+ Sol.
-# Module 5
+ Put your solution here.
-1. Give an example of two $$2-dim$$ subspaces in $$\mathbb{R^{3}}$$. Let us call it $$S_1, S_2$$.
+
-2. Let $$S_3$$ be all those vectors perpendicular to $$S_1$$. $$S_4$$ be that of $$S_2$$.
+49.What does $$S_1$$ and $$S_4$$ represent?
+
+ Sol.
+ Put your solution here.
-3. Find a matrix $$M$$ whose Null-Space is $$S_3$$. column space is $$S_2$$.
+
+50.Do you observe there is a bijection from $$S_1 \to S_2$$?
+
+ Sol.
-4. What does $$S_1$$ and $$S_4$$ represent?
+ Put your solution here.
-5. Do you observe there is a bijection from $$S_1 \to S_2$$?
+