Question: The system of equations at right 4x + 2y - 3z = 18 -2x + 2y - z = 2 3x + 2z = 2 has been worked using the Gauss-Jordan technique to the matrix below. Get the next matrix and STOP. 1 1/2 -3/4 l 9/2 0 3 -5/2 l 11 0 -3/2...

Question: Perform each of the indicated matrix operations below. If an operation is not defined, explain why it cannot be done. (There are brackets around the the numbers below for each matrix) a) \(\left[ \begin{matrix} 3 & -1 & 5 \\ -2 & 4 & 6 \\ \end{matrix} \right]-\left[ \begin{matrix} 1 & -2 & 2...

Question: Della's Deli Shop operates at 2 locations - a big discount store and on Main Street. She sells three sandwiches - chicken, ham, and steak. At the discount store this week she sold 75 ham sandwiches and 30 chicken sandwiches. On Main Street she sold 20 chicken and 35 ham sandwiches. She sold 30...

Question: Recall that we use the symbol \(\mathbb{R}\left[ x \right]\) to mean the real vector space of all polynomials in x with real coefficients. Show that the set \[\left\{ p\left( x \right)\in \mathbb{R}\left[ x \right]:\,\,p\left( 1 \right)=0\text{ and }\deg \left( p\left( x \right) \right)\le 3 \right\}\] is a subspace of \(\mathbb{R}\left[ x \right]\). Is it...

Question: Find real numbers a, b, and c so that the span of \(\left( 1,2,1 \right)\) and \(\left( 3,2,-5 \right)\) is the subspace of \({{\mathbb{R}}^{3}}\) given by \[{{\left( a,b,c \right)}^{\bot }}=\left\{ \left( x,y,z \right)\in {{\mathbb{R}}^{3}}:\,\,ax+by+cz=0 \right\}\]

Question: Find all real numbers \({{k}_{1}},{{k}_{2}},{{k}_{3}}\) so that the function \({{k}_{1}}\log x+{{k}_{2}}\log \left( x+1 \right)+{{k}_{3}}\log \left( x+2 \right)\) is constantly zero for all positive x. You may want to differentiate first, to avoid carrying a lot of logarithms around in your calculations.

Question: Express \(2{{x}^{3}}-9{{x}^{2}}+16x-9\) as a linear combination of \(x-1\), \({{\left( x-1 \right)}^{2}}\), and \({{\left( x-1 \right)}^{3}}\). That is, find real numbers \({{k}_{1}},{{k}_{2}},{{k}_{3}}\) so that \[2{{x}^{3}}-9{{x}^{2}}+16x-9={{k}_{1}}\left( x-1 \right)+{{k}_{2}}{{\left( x-1 \right)}^{2}}+{{k}_{3}}{{\left( x-1 \right)}^{3}}\]

Question: Consider the vectors in \({{\mathbb{R}}^{4}}\) defined by: \[{{\mathbf{v}}_{1}}=\left( -1,0,1,2 \right),\,\,{{\mathbf{v}}_{2}}=\left( 3,4,-2,5 \right),\,\,\,{{\mathbf{v}}_{3}}=\left( 1,4,0,9 \right)\] Find the reduced row echelon form of the matrix which has these as its rows. What is its rank? Is \(\left\{ {{\mathbf{v}}_{1}},{{\mathbf{v}}_{2}},{{\mathbf{v}}_{3}} \right\}\) linearly independent? What is the dimension of \(span\left\{ {{\mathbf{v}}_{1}},{{\mathbf{v}}_{2}},{{\mathbf{v}}_{3}} \right\}\) ? Find a homogeneous linear system for...