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00:53, 2 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 10 (b)/Statement (Created page with "The matrix A and the vectors v1, v2 and v3 are the same as in the previous part. Let W be a 2-dimensional subspace of R 3 such that, whenever a vector x is in W, the vector Ax is also in W. Suppose a vector")
00:50, 2 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 10 (a)/Statement (Created page with "Let <math> A </math> be a <math> 3\times 3 </math> matrix, and let <math> S=\{{\bf{v}}_1, {\bf{v}}_2, {\bf{v}}_3} </math> be a linearly independent set in <math> \mathbb{R}^3 </math> such that <math display = "block"> A{\bf{v}}_1={\bf{v}}_1, A{\bf{v}}_2=-{\bf{v}}_2, A{\bf{v}}_3={\bf{0}}. </math> Let <math> {\bf{w}}={\bf{v}}_1+{\bf{v}}_2+{\bf{v}}_3 </math>. Show that <math> {\bf{v}}_1,\;{\bf{v}}_2 </math> and <math> {\bf{v}}_3 </math> are all in <math> {\rm{Span}}\{{\b...")
00:31, 2 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 09 (d)/Statement (Created page with "Suppose now that, as well as the movement between Delta and Surrey, people immigrate from elsewhere (Abbotsford). This changes the dynamical system so that")
00:31, 2 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 09 (c)/Statement (Created page with "In the long run, what will the population of Surrey be?")
00:29, 2 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 09 (b)/Statement (Created page with "Let <math> \lambda_1 </math> and <math> \lambda_2 </math> denote the two eigenvalues of the matrix A. Compute <math> \lambda_1 </math> and <math> \lambda_2 </math> and two associated eigenvectors <math> {\bf{x}}_1 </math> and <math> {\bf{x}}_2 </math>. Find some constants <math> a </math> and <math> b </math> such that: <math display = "block"> \begin{bmatrix} d_n\\s_n\end{bmatrix} = a\lambda_1^n{\bf{x}}_1+ b\lambda_2^n{\bf{x}}_2. </math>")
00:18, 2 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 09 (a)/Statement (Created page with "There are two cities, Delta and Surrey. Every year • 10% of the population of Delta migrates to Surrey, • 20% of the population of Surrey migrates to Delta. Assume there are no other changes to the population of Delta or Surrey. In 2019 it was estimated that the population of Delta was 100,000 people, and that of Surrey was 500,000 people. Let dn and sn respectively denote the population of Delta and Surrey n years from now.")
00:18, 2 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 08 (d)/Statement (Created page with "Calculate <math> (I_3-A)^{888} </math>.")
00:16, 2 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 08 (c)/Statement (Created page with "Write down a basis for the null space of <math> A </math>.")
00:13, 2 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 08 (b)/Statement (Created page with "The transformation <math> T: \mathbb{R}^3\to \mathbb{R}^3 </math> given by <math> T(x)={\rm{proj}}_W{\bf{x}} </math> is linear. Write down the matrix <math> A </math> of <math> T </math>.")
00:01, 2 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 08 (a)/Statement (Created page with "Let <math display = "block> W=Span\left\{\begin{bmatrix} 1\\1\\1\end{bmatrix};\;\begin{bmatrix}1\\1\\-2\end{bmatrix}\right\}. </math> The following has been calculated already: <math display = "block> proj_{W}\begin{bmatrix}1\\0\\0\end{bmatrix}=\begin{bmatrix}1/2\\1/2\\0\end{bmatrix}. </math>")
23:45, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 07 (c)/Statement (Created page with "Consider the solution you found in the previous part. Call this vector <math> {\bf{v}} </math>. What is the distance between <math> A{\bf{v}} </math> and <math> \begin{bmatrix} 11\\0\11\\0\end{bmatrix}? </math>")
23:39, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 07 (b)/Statement (Created page with "The system <math display = "block"> A=\begin{bmatrix} 1& 0&1\\0&1&-1\\0&2&1\\0&-1&0\end{bmatrx},\quad {\bf{b}}=\begin{bmatrix} 11\\0\\11\\0\end{bmatrix} </math> is inconsistent. Find the least-squares solution of the system <math> A{\bf{x}}={\bf{b}}.</math>")
23:25, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 07 (a)/Statement (Created page with "Suppose A is an n × m matrix where n > m. Indicate the correct way to complete the sentence.")
23:23, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 06 (c)/Statement (Created page with "Let <math> V </math> be a subspace of <math> \mathbb{R}^7 </math> and let <math> {\bf{x}} </math> be a vector in <math> \mathbb{R}^7. </math> Suppose <math display = "block"> {\bf{v}}_1+ {\bf{u}}_1={\bf{x}}={\bf{v}}_2+{\bf{u}}_2, </math> where <math>{\bf{v}}_1,{\bf{v}}_2 </math> are in <math> V </math> and <math>{\bf{u}}_1,{\bf{u}}_2 </math> are in <math> V^{\perp} </math>. Show that <math>{\bf{v}}_1={\bf{v}}_2. </math>")
23:16, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 06 (b)/Statement (Created page with "Find a basis of the orthogonal complement <math> W^{\perp} </math>.")
23:14, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 06 (a)/Statement (Created page with "Find the dimension of the subspace <math> W </math> in <math> \mathbb{R}^5 </math> spanned by the following three vectors <math display = "block"> {\bf{v}}_1=\begin{bmatrix} 3\\1\\1\\0\\0\end{bmatrix},\;{\bf{v}}_2=\begin{bmatrix} -1\\2\\1\\0\\0\end{bmatrix},\;{\bf{v}}_3=\begin{bmatrix} -1/2\\-2\\7/2\\0\\0\end{bmatrix}.")
23:02, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 05 (c)/Statement (Created page with "Let <math display = "block"> A=\begin{bmatrix} 4&0&2\\ 0&3&0\\-1&0&1\end{bmatrix} </math>")
23:02, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 05 (b)/Statement (Created page with "Let <math display = "block"> A=\begin{bmatrix} 4&0&2\\ 0&3&0\\-1&0&1\end{bmatrix} </math>")
23:02, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 05 (a)/Statement (Created page with "Let <math display = "block"> A=\begin{bmatrix} 4&0&2\\ 0&3&0\\-1&0&1\end{bmatrix} </math>")
22:58, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 04 (e)/Statement (Created page with "Suppose <math> T: \mathbb{R}^3\to\mathbb{R}^2 </math> is a linear transformation such that <math display = "block"> T\left(\begin{bmatrix}1\\1\\1\end{bmatrix}\right)=\begin{bmatrix}1\\1\end{bmatrix},\;T\left(\begin{bmatrix}1\\0\\-1\end{bmatrix}\right)=\begin{bmatrix}2\\1\end{bmatrix}. </math> What is <math> T\left(\begin{bmatrix}1\\3\\5\end{bmatrix}\right)? </math>")
22:49, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 04 (d)/Statement (Created page with "In each of the following, indicate the correct answer (and only the correct answer) by circling the appropriate letter. You do not have to show your work in this question.")
22:43, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 04 (c)/Statement (Created page with "Consider the following statements about an n × n matrix A: 1: If the map x → Ax is one-to-one then it is also onto R n . 2: If the map x → Ax is onto R n then it is also one-to-one. Which of the statements above are true? A: Both statements are true B: Only statement 1 is true C: Only statement 2 is true D: Both statements are false")
22:40, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 04 (b)/Statement (Created page with "In each of the following, indicate the correct answer (and only the correct answer) by circling the appropriate letter. You do not have to show your work in this question. Suppose <math> A </math> is a <math> 2\times 2 </math> matrix <math> A=\begin{bmatrix}-1&1\\0&1\end{bmatrix}. </math> What is <math> A^{2019}? </math> '''<big>A:</big>''' <math> \;\begin{bmatrix} -1&2019\\0&1\end{bmatrix}\quad </math> '''<big>B:</big>''' <math>\; \begin{bmatrix} 2019&1\\0&2019\end{b...")
22:28, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 04 (a)/Statement (Created page with "In each of the following, indicate the correct answer (and only the correct answer) by circling the appropriate letter. You do not have to show your work in this question.")
22:27, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 03 (d)/Statement (Created page with "Let <math> A </math> be an <math> n\times n </math> matrix <math display = "block"> A=\begin{bmatrix} 0&1&0&0&\cdots&0&0\\ 0&0&1&0&\cdots&0&0\\ 0&0&0&1&\cdots&0&0\\ \vdots & \vdots & \vdots & \vdots & \ddots&\vdots & \vdots \\ 0&0&0&0&\cdots&1&0\\ 0&0&0&0&\cdots&0&1\\ 1&0&0&0&\cdots&0&0 \end{bmatrix} </math> Is this matrix invertible? If it is, find the inverse. If it is not, explain why not.")
22:18, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 03 (c)/Statement (Created page with "Let <math> A </math> be an <math> m\times m </math> invertible matrix. Moreover let <math> B </math> and <math> C </math> be <math> n\times m</math> matrices that satisfy <math= "block"> (B-C)A=0. </math> Show that <math> B=C. </math>")
22:15, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 03 (b)/Statement (Created page with "Is <math display = "block"> A=\begin{bmatrix} 0&1&2&3\\1&0&2&4\\ 0&2&1&3\end{bmatrix} </math> an invertible matrix? Give a clear justification of your answer (no credit will be given for an answer without a justification).")
22:13, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 03 (a)/Statement (Created page with "Write down the formula for the inverse of an invertible <math> 2\times 2 </math> matrix: <math> \begin{bmatrix} a& b\\c&d\end{bmatrix}.</math>")
22:02, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 02 (e)/Statement (Created page with "In each case, either give an example or state that no such example exists (you don’t have to explain why not). The matrix <math> A </math> of a linear transformation <math> T: \mathbb{R}^2\to\mathbb{R}^2, </math> different from the identity, such that the two sets <math display = "box"> \left\{ \begin{bmatrix} 0\\2 \end{bmatrix}, \; \begin{bmatrix} 1\\0 \end{bmatrix} \right\} </math> are equal.")
21:57, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 02 (d)/Statement (Created page with "In each case, either give an example or state that no such example exists (you don’t have to explain why not). A <math> 3 \times 3 </math> matrix <math> A </math> of real numbers such that <math> A </math> has no real eigenvalues.")
21:56, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 02 (c)/Statement (Created page with "In each case, either give an example or state that no such example exists (you don’t have to explain why not). A matrix <math> A </math> and a vector <math> \bf{b} </math> such that the homogeneous system <math> A\bf{x}=0 </math> has a unique solution, but such that <math> A\bf{x}=\bf{b} </math> has infinitely many solutions.")
21:53, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 02 (b)/Statement (Created page with "In each case, either give an example or state that no such example exists (you don’t have to explain why not). A square matrix <math> A </math> such that <math> det(A)\neq det(-A) </math>.")
21:52, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 02 (a)/Statement (Created page with "In each case, either give an example or state that no such example exists (you don’t have to explain why not). A subspace <math> W </math> of <math> \mathbb{R}^3 </math> and two vectors <math> \bf{v} </math> and <math> \bf{w} </math>, both in <math> W </math>, such that <math> \mathbf{v-w} </math> is not in <math> W. </math>")
21:48, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 01 (c)/Statement (Created page with "A common mistaken belief is that a system of linear equations with more variables than equations must have infinitely many solutions. Write down a system that shows that this belief is mistaken.")
21:46, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 01 (b)/Statement (Created page with "Consider the following linear system <math display="block"> \begin{align} -6x_1 + 6x_2 =& -12\\ x_1 - x_2 = &13\\ ax_1-bx_2=&3 \end{align} </math> Find values <math> a,\;b, </math> if they exist, such that the system is consistent. If such values do not exist, specify why they cannot exist.")
21:38, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 01 (a)/Statement (Created page with "Consider the linear system <math> \begin{align} -6x_1 + 6x_2 -11x_3-3x_4 =& -12\\ 2x_1 - 2x_2 + 2x_3-4x_4 = &14\\ x_1 - x_2 -5x_4 =& 13\\ 3x_1-3x_2+6x_3+3x_4=&3 \end{align} </math> State whether this system is consistent or inconsistent. If it is consistent, write down the set of solutions in parametric vector form. If it is inconsistent, explain how you can tell.")
21:20, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 10 (b) (Created page with "     Category:MER CQ flag Category:MER CH flag Category:MER CS flag Category:MER CT flag   <!-- To see the...")
21:20, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 10 (a) (Created page with "     Category:MER CQ flag Category:MER CH flag Category:MER CS flag Category:MER CT flag   <!-- To see the...")
21:19, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 09 (d) (Created page with "     Category:MER CQ flag Category:MER CH flag Category:MER CS flag Category:MER CT flag   <!-- To see the...")
21:19, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 09 (c) (Created page with "     Category:MER CQ flag Category:MER CH flag Category:MER CS flag Category:MER CT flag   <!-- To see the...")
21:19, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 09 (b) (Created page with "     Category:MER CQ flag Category:MER CH flag Category:MER CS flag Category:MER CT flag   <!-- To see the...")
21:19, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 09 (a) (Created page with "     Category:MER CQ flag Category:MER CH flag Category:MER CS flag Category:MER CT flag   <!-- To see the...")
21:18, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 08 (d) (Created page with "     Category:MER CQ flag Category:MER CH flag Category:MER CS flag Category:MER CT flag   <!-- To see the...")
21:18, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 08 (c) (Created page with "     Category:MER CQ flag Category:MER CH flag Category:MER CS flag Category:MER CT flag   <!-- To see the...")
21:18, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 08 (b) (Created page with "     Category:MER CQ flag Category:MER CH flag Category:MER CS flag Category:MER CT flag   <!-- To see the...")
21:18, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 08 (a) (Created page with "     Category:MER CQ flag Category:MER CH flag Category:MER CS flag Category:MER CT flag   <!-- To see the...")
21:17, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 07 (c) (Created page with "     Category:MER CQ flag Category:MER CH flag Category:MER CS flag Category:MER CT flag   <!-- To see the...")
21:17, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 07 (b) (Created page with "     Category:MER CQ flag Category:MER CH flag Category:MER CS flag Category:MER CT flag   <!-- To see the...")
21:17, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 07 (a) (Created page with "     Category:MER CQ flag Category:MER CH flag Category:MER CS flag Category:MER CT flag   <!-- To see the...")
21:16, 1 April 2025 AnaMucalica talk contribs created page Science:Math Exam Resources/Courses/MATH221/December 2019/Question 06 (c) (Created page with "     Category:MER CQ flag Category:MER CH flag Category:MER CS flag Category:MER CT flag   <!-- To see the...")