Students who are advanced or gifted tend to display a surpri…

Questions

Students whо аre аdvаnced оr gifted tend tо display a surprisingly homogeneous set of learning needs and characteristics.

C1.) [10 pоints] Cоnvert the fоllowing system of lineаr equаtions into mаtrix form, solve the system via row reduction, and state the solutions, using parameters if necessary. begin{align}w+2x-y+z&=5\w+y-2z&=3\w+6x-5y+7z&=9end{align} C2.) Let (A=begin{bmatrix}1&2&3\-3&-1&4\2&1&5end{bmatrix}) and (B=begin{bmatrix}1&-1&3\1&2&1\0&2&1end{bmatrix}). Find each of the following. a.) [10 points] (AB) b.) [10 points] the determinant of (B) C3.) [10 points] Use the Wronskian to show that the set (left{x,cosleft(xright),e^{x}right}) is linearly independent. Make sure you explain what evidence in your work substantiates your answer. C4.) [10 points] Determine whether or not the set (left{begin{bmatrix}1\1\1\3end{bmatrix},begin{bmatrix}1\-1\2\-2end{bmatrix},begin{bmatrix}0\0\1\4end{bmatrix}right}) is linearly independent. Make sure you explain what evidence in your work substantiates your answer. C5.) [10 points] Let (f=2-x+x^{2}). Find (left(fright)_{mathcal{B}}), where (mathcal{B}=left{1,1+x,1-x^{2}right}). C6.) [10 points] Let (fleft(xright)=sqrt{x}) and (gleft(xright)=x^{2}). Find (leftlangle f,grightrangle), using the integral inner product on (Cleft[0,1right]). C7.) [10 points] Given the following [A=begin{bmatrix}2&-6&1&5&-1\1&-3&-1&1&-5\-1&3&2&0&8\4&-12&-3&5&-17end{bmatrix}qquad text{rref}left(Aright)=begin{bmatrix}1&-3&0&2&-2\0&0&1&1&3\0&0&0&0&0\0&0&0&0&0end{bmatrix}] Find bases for the spaces: (text{row}left(Aright)), (text{col}left(Aright)), and (text{null}left(Aright)). Be sure to label which basis is which. C8.) [10 points] Let (A=begin{bmatrix}1&1\4&1end{bmatrix}). Find the eigenvalues for (A). You should find two distinct values, (lambda_1) and (lambda_2), with (lambda_1 < lambda_2). Then find a basis for the eigenspace corresponding to (lambda_1), the smaller eigenvalue. C9.) [10 points] Let (overrightarrow{u}=begin{bmatrix}1\-2\4end{bmatrix}) and (overrightarrow{v}=begin{bmatrix}0\5\-3end{bmatrix}) be vectors in (mathbb{R}^{3}), with a weighted Euclidean inner product given by the weights (w_{1}=3), (w_{2}=2), and (w_{3}=5). Find the angle between (overrightarrow{u}) and (overrightarrow{v}), to the nearest tenth of a degree. C10 [10 points] Let (overrightarrow{u}=begin{bmatrix}1\-2\4end{bmatrix}) and (overrightarrow{v}=begin{bmatrix}0\5\-3end{bmatrix}) be vectors in (mathbb{R}^{3}) once more, but now with the inner product generated by the matrix (A=begin{bmatrix}1&0&-1\0&1&-1\1&1&1end{bmatrix}). Find the projection of (overrightarrow{u}) onto (overrightarrow{v}). C11.) [10 points] Let (W) be the subspace of (mathbb{R}^{4}) with the following set as a basis [left{begin{bmatrix}1\2\2\-1end{bmatrix},begin{bmatrix}1\-1\0\0end{bmatrix},begin{bmatrix}1\2\1\1end{bmatrix}right}] Convert this basis to an orthogonal basis for (W). Do not convert it to an orthonormal basis. C12.) [10 points] Find the least squares solution to the following inconsistent system. You do NOT have to find the least squares error vector or error. begin{align}3x-y&=10\x+2y&=-2\-x+y&=1end{align} C13.) [10 points] Let (f) be the transformation on (mathbb{R}^{2}) that is reflection in (or across) the (y)-axis. Let (g) be the transformation on (mathbb{R}^{2}) that is rotation counterclockwise by (120^{circ}). Write down the standard matrix for each of these transformations.