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- 21:12, 15 April 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B5 (c)/Hint 2 (Created page with "The matlab command <math>[P,D]=\mathrm{eig}(A)</math> yields the matrix <math>P</math>, whose columns are the eigenvectors of <math>A</math>, and the diagonal matrix <math>D</math>, whose entries are the eigenvalues of <math>A</math>.")
- 21:14, 14 April 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question A18/Solution 2 (Created page with "We will follow the second hint. Let <math>v \in \mathbb{R}^2</math> be a vector that makes an angle of <math>30^\circ</math> with the <math>x</math>-axis. Then the first reflection fixed <marth>v</math>, so the composite <math>\mathit{Ref}_{50^\circ}\circ\mathit{Ref}_{30^\circ}</math> takes <math>v</math> into <math>\mathit{Ref}_{50^\circ}(v)</math>, which is a vector that makes an angle of 40º with <math>v</math>, in the counterclock-wise direction. Therefore, <math>\m...")
- 21:00, 14 April 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question A18/Hint 2 (Created page with "Here is a hack that allows for a quick solution just in this case. The question statement already tells us that the composite <math>\mathit{Ref}_{50^\circ} \circ \mathit{Ref}_{30^\circ}</math> is a rotation, so we just need to compute what it does to a single vector, for example a unit vector that is fixed by <math>\mathit{Ref}_{30^\circ}</math>.")
- 02:09, 9 April 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B3 (b)/Solution 1 (Created page with "We are looking for bases of the null-spaces of the matrices <math display="block"> \left[\begin{array}{ccc} 0 - \lambda & 5 & -5 \\ 1 &4-\lambda& 1 \\ 0 & 0 &5-\lambda \end{array}\right], </math> for <math> \lambda = -1, 5</math>. Since the eigenvalue 5 is repeated, this is the value of <math>\lambda</math> for which the matrix above has a 2-dimensional null-space. We apply row operations to find the null-spaces. For <math>\lambda = -1</ma...")
- 04:32, 5 April 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B3 (b)/Hint 1 (Created page with "Based on the work in part (a), we are looking for vector solutions to the equations <math display="block"> Av = -v \qquad \text{and} Av = 5v .</math> Each vector equation above is a system of three linear equations. Can you try to solve these? Which one do you expect to have a two-dimensional solution space?")
- 04:28, 5 April 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B3 (b)/Statement (Created page with "This is a continuation of [https://wiki.ubc.ca/Science:Math_Exam_Resources/Courses/MATH152/April_2022/Question_B3_(a)|B3 (a)]. (b) Find three linearly independent eigenvectors of <math>A</math>.")
- 04:26, 5 April 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B3 (a)/Solution 1 (Created page with "Here is the argument for the standard procedure that computes the eigenvalues of a matrix. The equation <math display="block"> Av = \lambda v</math> is equivalent to <math display="block"> (A-\lambda\mathrm{Id})v = 0.</math> This latter equation has a non-zero solution if and only if the matrix <math> A - \lambda\mathrm{Id}</math> is ''not'' invertible, which is equivalent to <math>\det(A-\lambda\mathrm{Id}) = 0</math>. Therefore, we solve this last equation, since it is...")
- 04:06, 5 April 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B3 (a)/Hint 1 (Created page with "Recall the definition: an eigenvalue of <math>A</math> is a number <math>\lambda</math> such that the (vector) equation <math display="block"> Av = \lambda v</math> has a solution. Can you rewrite the above equation so that the vector <math>v</math> is in the null-space of a matrix?")
- 04:03, 5 April 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B3 (a)/Statement (Created page with "Let <math display="block"> A = \left[\begin{array}{ccc} 0 & 5 & -5 \\ 1 & 4 & 1 \\ 0 & 0 & 5 \end{array}\right]. </math> (a) Find the eigenvalues of <math>A</math>.")
- 23:55, 4 April 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B2 (c)/Solution 1 (Created page with "Suppose that, in the long run, <math> v_1, v_2, v_3 </math> are the proportions of buyers who buy Ambrosia, Braeburn and Cameo apples, respectively. Then the transition matrix <math>A</math> should leave the vector <math>\left[\begin{array}{c} v_1 \\ v_2 \\ v_3 \end{array}\right] </math> fixed. In other words, we are looking for an eigenvector with eigenvalue 1, or an element of the null-space of the matrix <math display="block"> A - \mathrm{Id} = \left[\begin{array}{...")
- 23:29, 4 April 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B2 (c)/Hint 1 (Created page with "If the share of each apple stabilizes, how should the transition matrix act on the vector of shares?")
- 23:13, 4 April 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B2 (b)/Solution 1 (Created page with "There are 3 ways to walk from B to B in two steps: <math display="block"> B \rightarrow B \rightarrow B,\qquad B \rightarrow A \rightarrow B,\qquad B \rightarrow C \rightarrow B.</math> Each individual path has probability <math> p_{2,i} \cdot p_{i,2}</math>, for <math>i = 1,2,3</math>, and where <math>p_{i,j}</math> is the probability of going from vertex <math>j</math> to vertex <math>i</math> (which is also the <math>(i,j)</math> entry of the transition matrix comp...")
- 23:04, 4 April 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B2 (b)/Hint 1 (Created page with "The probability of that a Braeburn buyer will buy a Braeburn again in two weeks is the probability of starting at vertex B and ending up at vertex B again after two steps. How many walks are there that take two steps, start at B and end at B?")
- 22:59, 4 April 2024 MihaiMarian talk contribs uploaded File:Math Exam Resources-Courses-MATH152-April 2022-Question B2 (a) Hint.jpg (Uploaded own work with UploadWizard) Tag: Upload Wizard
- 22:59, 4 April 2024 MihaiMarian talk contribs created page File:Math Exam Resources-Courses-MATH152-April 2022-Question B2 (a) Hint.jpg (Uploaded own work with UploadWizard)
- 02:33, 3 April 2024 MihaiMarian talk contribs moved page Science:Math Exam Resources/Courses/MATH100 C/December 2023/Question 05/Hint 1 to Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 05/Hint 1 over redirect (Undoing a move)
- 02:33, 3 April 2024 MihaiMarian talk contribs deleted redirect Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 05/Solution 1 by overwriting (Deleted to make way for move from "Science:Math Exam Resources/Courses/MATH100 C/December 2023/Question 05/Solution 1")
- 02:33, 3 April 2024 MihaiMarian talk contribs moved page Science:Math Exam Resources/Courses/MATH100 C/December 2023/Question 05/Solution 1 to Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 05/Solution 1 over redirect (Undoing a move)
- 02:33, 3 April 2024 MihaiMarian talk contribs deleted redirect Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 05/Statement by overwriting (Deleted to make way for move from "Science:Math Exam Resources/Courses/MATH100 C/December 2023/Question 05/Statement")
- 02:33, 3 April 2024 MihaiMarian talk contribs moved page Science:Math Exam Resources/Courses/MATH100 C/December 2023/Question 05/Statement to Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 05/Statement over redirect (Undoing a move)
- 02:33, 3 April 2024 MihaiMarian talk contribs moved page Science:Math Exam Resources/Courses/MATH100 C/December 2023/Question 05 to Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 05 over redirect (Undoing a move)
- 02:33, 3 April 2024 MihaiMarian talk contribs deleted redirect Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 05/Hint 1 by overwriting (Deleted to make way for move from "Science:Math Exam Resources/Courses/MATH100 C/December 2023/Question 05/Hint 1")
- 02:33, 3 April 2024 MihaiMarian talk contribs deleted redirect Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 05 by overwriting (Deleted to make way for move from "Science:Math Exam Resources/Courses/MATH100 C/December 2023/Question 05")
- 02:29, 3 April 2024 MihaiMarian talk contribs moved page Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 05/Statement to Science:Math Exam Resources/Courses/MATH100 C/December 2023/Question 05/Statement (Attempting to copy this page to the C flavour, since it is the same question)
- 02:29, 3 April 2024 MihaiMarian talk contribs moved page Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 05 to Science:Math Exam Resources/Courses/MATH100 C/December 2023/Question 05 (Attempting to copy this page to the C flavour, since it is the same question)
- 02:29, 3 April 2024 MihaiMarian talk contribs moved page Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 05/Hint 1 to Science:Math Exam Resources/Courses/MATH100 C/December 2023/Question 05/Hint 1 (Attempting to copy this page to the C flavour, since it is the same question)
- 02:29, 3 April 2024 MihaiMarian talk contribs moved page Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 05/Solution 1 to Science:Math Exam Resources/Courses/MATH100 C/December 2023/Question 05/Solution 1 (Attempting to copy this page to the C flavour, since it is the same question)
- 07:14, 28 March 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B2 (a)/Solution 1 (Created page with "According to the hint, the transition matrix has <math>ij</math>-entry equal to the probability of going from state <math> j </math> to state <math> i </math> (where states A, B, C are thought of as labelled by the indices 1, 2, 3, respectively): <math display="block"> \left[\begin{array}{ccc} 0.7 & 0.2 & 0.2 \\ 0.2 & 0.6 & 0.4 \\ 0.1 & 0.2 & 0.4 \end{array}\right]. </math>")
- 06:56, 28 March 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B2 (a)/Hint 1 (Created page with "It may be helpful model the random walk using a graph with vertices labelled <math> A, B, \text{ or } C </math>, for the events <math>A=</math> "buying Ambrosia," etc. and with edges labelled by the probabilities of moving from one event to the other over the course of a week. Recall that it is these probabilities that are the inputs of the transition matrix.")
- 04:31, 26 March 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B2 (c)/Statement (Created page with "This is a continuation of [https://wiki.ubc.ca/Science:Math_Exam_Resources/Courses/MATH152/April_2022/Question_B2_(a)|Question B2 (a)]. (c) In the long run, what will the share of each type of apple be?")
- 04:31, 26 March 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B2 (b)/Statement (Created page with "This is a continuation of [https://wiki.ubc.ca/Science:Math_Exam_Resources/Courses/MATH152/April_2022/Question_B2_(a)|Question B2 (a)]. (b) Find the probability that a Braeburn buyer will buy Braeburn again after two weeks.")
- 04:30, 26 March 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B2 (a)/Statement (Created page with "A grocery store sells three types of apples: Ambrosia, Braeburn and Cameo. Every week, 20% of the people who bought Ambrosia last week switch to Braeburn, 10% switch to Cameo, and 70% buy Ambrosia again. Similarly, 20% of customers who bought Braeburn last week switch to Ambrosia, 20% switch to Cameo and the remaining 60% buy Braeburn again. Those who bought Cameo last week, 20% switch to Ambrosia, 40% switch to Braeburn, and 40% stay with Cameo. (a) Find the transition...")
- 04:28, 26 March 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B1 (a)/Hint 2 (Created page with "Let <math> n_A,\ n_B,\ n_C,\ n_D </math> denote the numbers of boxes of each type of cookie that were produced. Then we have the following system of equations, where each row corresponds to the use of one of the ingredients: <math display="block"> \begin{cases} n_A + 2n_B + n_C + 3n_D &= 11\\ 3n_A+ 6n_B +3n_C + 9n_D &= 33\\ n_A + 4n_B +3n_C + 7n_D &= 21\\ 2n_A+ n_B + 3n_C+ 8n_D &= 23\\ 5n_B + n_C + 2n_D &= H \end{cases}. </math> Did you learn tools to reduce su...")
- 04:27, 26 March 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B1 (b)/Hint 2 (Created page with "Let <math> n_A,\ n_B,\ n_C,\ n_D </math> denote the numbers of boxes of each type of cookie that were produced. Then we have the following system of equations, where each row corresponds to the use of one of the ingredients: <math display="block"> \begin{cases} n_A + 2n_B + n_C + 3n_D &= 11\\ 3n_A+ 6n_B +3n_C + 9n_D &= 33\\ n_A + 4n_B +3n_C + 7n_D &= 21\\ 2n_A+ n_B + 3n_C+ 8n_D &= 23\\ 5n_B + n_C + 2n_D &= H \end{cases}. </math> Did you learn tools to reduce su...")
- 04:22, 26 March 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B1 (b)/Solution 1 (Created page with "We have the following reduced augmented matrix: <math display="block"> \left[\begin{array}{cccc|c} 1 & 0 & 0 & 1 & 5 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 2 & 4 \\ 0 & 0 & 0 & 0 & H-9 \end{array}\right], </math> which corresponds to the system of equations <math display="block"> \begin{cases} n_A + n_D &= 5\\ n_B &= 1\\ n_C + 2n_D &=4 \end{cases}. </math> We see that there can be multiple solutions, but that the system can be solved in ter...")
- 04:14, 26 March 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B1 (b)/Hint 1 (Created page with "In [https://wiki.ubc.ca/Science:Math_Exam_Resources/Courses/MATH152/April_2022/Question_B1_(a)|Question B1 (a)], we put the augmented matrix describing the mathematical problem in a special reduced form. Does the linear system look like it has a unique or multiple solutions?")
- 04:10, 26 March 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B1 (b)/Statement (Created page with "This is a continuation from Question B1 (a). (b) Find how many boxes of each type were produced. Is there more than one answer? If so, give the general solution of the mathematical problem in parametric form. Note: it is '''not''' necessary to ensure that the solution consist of positive integers.")
- 04:05, 26 March 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B1 (a)/Solution 1 (Created page with "The linear system described in the hint is given by the following augmented matrix: <math display="block"> \left[\begin{array}{cccc|c} 1 & 2 & 1 & 3 & 11 \\ 3 & 6 & 3 & 9 & 33 \\ 1 & 4 & 3 & 7 & 21 \\ 2 & 1 & 3 & 8 & 23 \\ 0 & 5 & 1 & 2 & H \end{array}\right] </math> We can solve the linear system by applying row operations: <math display="block"> \left[\begin{array}{cccc|c} 1 & 2 & 1 & 3 & 11 \\ 3 & 6 & 3 & 9 & 33 \\ 1 & 4 & 3 & 7 & 21 \\ 2 & 1 & 3 & 8 & 23 \\ 0 & 5...")
- 03:49, 26 March 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B1 (a)/Hint 1 (Created page with "A number of boxes of each type of cookie was produced. Use four variables for these numbers of boxes and write down a linear system describing the total amount of ingredients used as a function of these 4 variables. Can you express this system as an augmented matrix to solve for <math> H </math>?")
- 03:45, 26 March 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question B1 (a)/Statement (Created page with "A factory produces four types of cookies: A, B, C and D. Ingredients needed to make one box of cookies of each type are given in the table below: {| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;" |- | || A || B || C || D |- | butter (lb.) || 1 || 2 || 1 || 3 |- | flour (cups) || 3 || 6 || 3 || 9 |- | eggs || 1 || 4 || 3 || 7 |- | sugar (cups) || 2 || 1 || 3 || 8 |- | chocolate (lb.) || 0 || 5 || 1 || 2 |...")
- 03:02, 25 March 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question A10/Solution 1 (Created page with "Following the hint, we simply need to find the result of the linear transformation on each basis vector: <math display="block">\begin{align} T\left(\left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right]\right) &= \left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right] \times \left[\begin{array}{c} 0 \\ 1 \\ 0 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right],\\ T\left(\left[\begin{array}{c} 0 \\ 1 \\ 0 \end{array}\right]\right) &= \left[\begin{arra...")
- 02:57, 25 March 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question A10/Hint 1 (Created page with "Recall that the columns of a matrix are the result of multiplying the basis vectors by the matrix. For example, <math display="block"> \left[\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{33} \\ a_{31} & a_{32} & a_{33} \end{array}\right] \cdot \left[\begin{array}{c} 0 \\ 1 \\ 0 \end{array}\right] = \left[\begin{array}{c} a_{12} \\ a_{22} \\ a_{32} \end{array}\right]. </math>")
- 02:52, 25 March 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question A10/Statement (Created page with "Find the matrix of the linear transformation <math> T \colon \mathbb{R}^3 \rightarrow \mathbb{R}^3 </math> given by <math display="block"> T(\vec{x}) = \vec{x} \times \left[\begin{array}{c} 0 \\ 1 \\ 0 \end{array}\right]. </math> Here <math> \times </math> denotes the cross product.")
- 02:48, 25 March 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question A09/Solution 1 (Created page with "The planes defining the line have normal vectors <math> \vec{v} = \left[\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right] \text{ and } \vec{w} = \left[\begin{array}{c} 1 \\ 1 \\ 1 \end{array}\right] </math>. Their cross-product yields a vector that is parallel to <math> L </math>: <math display = "block"> \vec{v} \times \vec{w} = \left[\begin{array}{c} -1 \\ 2 \\ -1 \end{array}\right]. </math> For this latter vector to be parallel to the plane given by <math> 2x + cz = 2...")
- 02:35, 25 March 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question A09/Hint 1 (Created page with "The line <math> L </math> is given as an intersection of two planes. What relationship is there between the three planes' normal vectors?")
- 02:33, 25 March 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question A09/Statement (Created page with "Find a constant <math> c </math> such that the plane defined by <math> 2x + cz = 2 </math> is parallel to the line <math display="block"> L: \begin{cases} x + 2y + 3z = 3 \\ x + y + z = 4 \end{cases}. </math>")
- 02:28, 25 March 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question A08/Solution 2 (Created page with "Following the second hint, consider the three vectors from <math> (1,-1,2) </math> to the points <math> (1,1,0),\ (2, -1, 1),\ \text{ and } (0, 1, c) </math>. That these three vectors are linearly dependent is characterized by the fact that the parallelepiped that they span has no volume or, equivalently that the matrix formed by these three vectors has determinant equal to 0: <math display="block"> 0 = \det\left[\begin{array}{ccc} 0 & 2 & -2 \\ 1 & 0 & -1 \\ -1& 2 & c-...")
- 02:15, 25 March 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question A08/Hint 2 (Created page with "For a more slick solution, note that if four points are co-planar, then the three vectors that go from one point to each of the other three must necessarily be linearly dependent.")
- 02:13, 25 March 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question A08/Solution 1 (Created page with "We will use the following approach. Note that the difference between two points on a plane is a vector that is parallel to the plane. If we can find two such vectors (that are not co-linear), then their cross-product is normal to the plane and can be used to write down an equation that the points on the plane need to satisfy. Let then <math display="block"> \begin{align} \vec{v} &= (2,-1,1) - (1,1,0) = &\left[\begin{array}{c} 1 \\ -2 \\ 1 \end{array}\right]\\ \vec{w} &...")
- 00:43, 25 March 2024 MihaiMarian talk contribs created page Science:Math Exam Resources/Courses/MATH152/April 2022/Question A08/Hint 1 (Created page with "Can you write down an equation that the points on the plane need to satisfy? Note that it will depend on <math> c </math>.")