Mathematics 1229A/B Study Guide - Midterm Guide: Dot Product, Parallelepiped, Wicket-Keeper

The characteristic equation of the matrix A = [5 4 1 -3] is a. lambda^2 - 2 lambda - 19 = 0 b. lambda^2 + 2 lambda - 19 = 0 c. lambda^2 - 2 lambda + 19 = 0 d. lambda^2 + 2 lambda + 19 = 0 Cosine value of the angle between vectors u and v is 1 and ||u|| = 3, ||v|| = 5, then (u, v) a. 10 b. 15 c. Squareroot 15 d. Squareroot 10

TRUE-FALSE, Determine whether each statement below is either true of false. Write either TRUE or FALSE (all caps), as appropriate, before each one. a) ____ A linear system with a reduced row echelon form having a row of zeroes has no solutions. b) ____ No set of 6 dependent vectors in R^5 spans R^5. c) ____ Let V be a vector space and S be a subset of V. If V = span(S), then s is a basis for V. d) _____ If u middot v = 0 then either u = 0 or v = 0. e) _____ If v is a 5-component vector of norm 4, then ||3v|| = (3)^5 4 f) _____ The solution set of a consistent nonhomogeneous linear sysem Ax = b, where A is an m times n matrix and b is m times 1, is a subspace of R^n g) ____ A linear system with a reduced row echelon form having a row of zeroes has infinitely many solutions. h) _____ If u, v and w are all 5-component vectors then (u middot v) middot w = u middot (v middot w) h) ____ If u, v and w are all 5-component vectors then (u middot v) middot w = u middot (v middot w) i) ____ The general solution of the nonhomogeneous linear system Ax = b can be obtained by adding b to the general solution of the homogeneous linear system Ax = 0.

Which of the following is the LU-factorization of the matrix [1 -4 0 1]? A) [1 0 -4 1][1 0 0 -1] B) [1 0 1/4 1][1 0 0 1] C) [1 -1/4 0 1][1 0 0 -1] D) [1 -4 0 1][1 0 0 1] E) [1 4 0 1][1 0 0 -1] The eigenvalues of the matrix [4 8 3 -1] are lambda_1 = -4 and lambda_2 = 7. Find an eigenvector corresponding to each of the given eigenvalues. Find the eigenvalues of the matrix [6 -9 -4 1] Use Cramer's Rule to solve the system of equations {8x-7y = 7 3x-y = 3. Find the area of the triangle having the vertices (-5, 3), (4, 6), and (-4, 6). Find the vector v for which v=u+6w given that u=(5, 7) and w=(-6, -5). Provided that u=(5, 4) and w=(2, -4), write v=(31, 8) as a linear combination of u and w, if possible.