Prove that the matrix A = [0 1 1 0] has no LU factorization, i. e., no lower triangular matrix L and upper triangular matrix U exist, such that A = LU. Hint: Assume that there is an LU decomposition [l11 0 l21 l22] [u11 u12 0 u22] = [0 1 1 0] and arrive at a contradiction!
Suppose that a 3×3 matrix A factors into a product (1 0 0 l21 1 0 l31 l32 1)(u11 u12 u13 0 u22 u23 0 0 u33) Determine the value of det(A).
Suppose that a 3×3 matrix A factors into a product: A = LU = [1 0 0 l21 1 0 l31 l32 1][13 u12 u13 0 1 u23 0 0 1/13] Determine the value of det(A).
a). Compute the LU factorization of: A = [2 4 2 1 1 2 −1 0 2] b). Use the LU factorization to solve the system: Ax = [2 1 4], where x = [x1 x2 x3]
The truncated conical container shown is full of a beverage that weighs 0.55 oz/in3. The container is 9 in. deep, 2.8 in. across at the base, and 4 in. across at the top. A straw sticks up 3 in. above the top. How much work does it take to suck up the beverage through the straw (neglecting friction)? How much work is required? in-oz. (Round to the nearest tenth as needed.)
Find a vector equation and parametric equations for the line. (Use the parameter t.) the line through the point (3, 2, 4) and parallel to the vector 4i − j + 5k r(t) = (x(t), y(t), z(t)) = ( )
Find a vector equation and parametric equations for the line. (Use the parameter t.) the line through the point (7, 0, −3) and parallel to the line x = 4 − 2t, y = −1 + 3t, z = 6 + 8t r(t) = (x(t), y(t), z(t)) = ( )
Find a vector equation and parametric equations for the line. (Use the parameter t.) the line through the point (4, 9, 1) and perpendicular to the plane 4x − 4y + 4z = 5 (x(t), y(t), z(t)) = ( )
Find parametric equations for the line. (Use the parameter t.) the line through the origin and the point (6, 5, −1) (x(t), y(t), z(t)) = ( ) Find the symmetric equations. x + 6 = y + 5 = z − 1 x/6 = y/5 = z x/6 = y/5 = −z x − 6 = y − 5 = z + 1 x/5 = y/6 = −z
Consider the line through the points (−4, −6, 1) and (−2, 0, −3). Write a direction vector v1 for this line. v1 = Consider the line through the points (7, 12, 13) and (4, 3, 19). Write a direction vector v2 for this line. v2 = Are v1 and v2 parallel? Yes No Is the line through (−4, −6, 1) and (−2, 0, −3) parallel to the line through (7, 12, 13) and (4, 3, 19)? Yes No
Is the line through (−3, 3, 0) and (1, 1, 1) perpendicular to the line through (2, 3, 4) and (6, −1, −7)? For the direction vectors of the lines, v1⋅v2 = Therefore, the lines perpendicular.
Determine whether the lines L1 and L2 are parallel, skew, or intersecting. L1: x = 8 − 12t, y = 4 + 9t, z = 9 − 3t L2: x = 4 + 8s, y = −6s, z = 9 + 2s parallel skew intersecting If they intersect, find the point of intersection. (If an answer does not exist, enter DNE.) (x, y, z) = ( )
Find an equation of the plane. the plane through the point (8, 2, 6) and with normal vector 3i + 3j + 3k
Find an equation of the plane. the plane through the point (8, −3, −4) and parallel to the plane z = 3x − 4y
Find an equation of the plane. the plane through the points (2, 1, 2), (3, −8, 6), and (−2, −3, 1)
Use intercepts to help sketch the plane. 5x + 2y + z = 10 x-intercept x = y-intercept y = z-intercept z =
Determine whether the planes are parallel, perpendicular, or neither. 5x + 20y − 15z = 1, −24x + 48y + 56z = 0 parallel perpendicular neither If neither, find the angle between them. (Use degrees and round to one decimal place. If the planes are parallel or perpendicular, enter PARALLEL or PERPENDICULAR, respectively.)