Math Test Review. Given two points in R, x, y, z and x, y, z, show the point x + x, y + y, z + z is on the line between these two points and is the same distance from each of them. The line is rt x, y, z +t x, y, z x, y, z x +t x x, y +t y y, z +t z z. Now let t / and you get the desired point.. Given the two points in R, x, y, z and x, y, z, describe the set of all points which are equidistant from these two points in terms of a simple equation. Let x x + y y + z z x x + y y + z z. Square both sides and cancel things out: x xx + x + y yy + y + z zz + z x xx + x + y yy + y + z zz + z. Then the equation of the plane is x x x + y y y + z z z x + y + z x + y + z. n airplane heads due north at a speed of miles per hour. The wind is blowing East at a speed of miles per hour. Find the resulting speed of the airplane. The velocity of the plane is j and the velocity of the wind is i Then the resultant velocity is i + j and so the speed is + 7. Find the cosine of the angle between the two vectors i + j and i + k. cos θ,,,, 5 5 5 5. Suppose u, v are vectors. Show the parallelogram identity. u + v + u v u + v You must show this in any dimension, not just in two or three dimensions. u + v + u v u + v u + v+u v u v u + v, which results after simplifying. 6. Let u, v be vectors in R n show using the properties of the dot product that u v u + v u v. Using the properties of the dot product the right side is u + u v + v u u v + v u v 7. Find proj u v, the projection of v i + j + k on u i + j + k. v u,,,, proj u v u,, u u + 9 +,,. 8. Given two vectors, u, v in R n, show using the properties of the dot product alone that is perpendicular to v. u u v v v u u v v v v u v u v v v v u v u v. 9. Find the line between the two points,, and,,. Write the vector parametrization, the scalar parametric equations, and the symmetric form. The vector parameteriztion is rt,, +t,,,, t,, t ti+j+ tk. The parametric equations are x t, y and z t. The symmetric form is x z, y.
. Let a i + j + k and b i j + k. Find a vector which is perpendicular to both of these vectors. a b 5i + 5j 5k, or any other vector in the same direction. Find the area of the parallelogram determined by a i + j + k and b i j + k. a b 5i + 5j 5k 5 + 5 + 5 5. Find the cosine of the angle between a i + j + k and b i j + k. cos θ a b a b,,,, ++9 ++ 6. Find the sine of the angle between a i + j + k and b i j + k. sin θ a b a b,,,, 5,5, 5 ++9 ++ ++9 ++ 5 6 5 7. Find the volume of the parallelepiped determined by the vectors, a i + j + k, b i j + k and c i + j + k. Volume a b c,,,,,, 5 5 5. plane has a normal vector n i k and contains the point,,. Find the equation of the plane. x + y + z, or in other words x z 5. 6. Find the equation of a plane which contains the three points,,,,,, and,,. First you find the normal.,,,,,,,,,,. Now with this normal, the equation of the plane is x + y + z, or in other words x + y z. 7. Find the cross product, i j + k i + k. What is the area of the triangle determined by these two vectors. rea,,,,,, + 8. Find the equation of a plane which is parallel to the plane whose equations is x + y + z 7 which contains the point,,. x + y + z +, or in other words x + y + z 6. 9. Where does the line rt + ti + + tj + + tk intersect the plane x + y + z? x + y + z + t + + t + + t. Solve for t to get t 5 6. Now plug in this and get r 5 6 + 5 6 i + + 5 6 j + 5 6 k i + j + 6 k. Thus the point is,, 6.. Find the volume of the parallelepiped determined by the vectors a i + j + k, b i + j, and c i + k. Volume a b c,,,,,,. Here is a system of equations. Find the complete solution. x + y + z x + y + z y + z The solution is x t, y t +, z t where t R.. Multiply the matrices, if possible. a b c a 5 b c. True or False. In each case the capital letters are matrices of an appropriate size and the lower case letters represent scalars. If a statement is false, give an example which shows the statement is false or else explain why it makes no sense. a B B + B
b B T T B T c a + bb C ac + bcb d If B, then either or B. e / f B C BC a False b False c False d False. e False. f True. Find its inverse of 5. Write the solution to the system in terms of a, b, c given that x a b, y a b c, z a + b + c 6. Find the solution space to x for We say the solutions are x x + y + z a y + z b x y c if, then. s t s t s t 7. Let 7 8. or, in other words, the solution space is Span a b a a a b be the augmented matrix for a linear system. Find for what values of a and b the system has a a unique solution b a one-parameter solution c a two-parameter solution d no solution The matrix reduces to a b a b. Thus b b a There is a unique solution when a and b. b There is a one-parameter solution when a and b.,.
c There is a two-parameter solution when a and b,. d There is no solution when a and b,. 8. Identify the pivot columns the columns with leading s in the rref of the matrix R R R R R R R R R pivot positions. Therefore the pivot columns are the first and second columns of.. Thus a and a are the 9. What matrix will produce the row operation of taking times the first row and adding it to the third row when multiplied on the left of a 5 matrix?. Find the inverse of the matrix using a row reduction and b the adjoint formula. a Reduce [ I] to [I ]. b Use det adj det [C ij] T. In both cases,. Solve the matrix equation below for X. X X We can rewrite X B X as X X B. Then IX B. Solving for X we obtain X I B. Hence X. Solve the system x b using the given LU factorization of : ns. x. Find the LU factorization of the matrix: 5 5, b Recall this is a factorization in which L has ones down the main diagonal and is lower triangular while U is upper triangular. ns.
. If is an invertible matrix, show T is also an invertible matrix and that T T. Use the definition: matrix X has inverse Y if and only if XY I and Y X I. In this case, T takes on the role of X and T takes on the role of Y. To see that T is the inverse of T, observe that T T T I T I and T T T I T I. Therefore T T. Make sure you include the justification of each step in your proof. 5. Determine whether the vectors below are dependent or independent. If they are dependent, exhibit one of them as a linear combination of the others. Solve c + + + c + c. Hence,, to get c t, c t, and c t. Let t. Then 6. linear transformation involves first rotating the vectors in R counterclockwise through an angle of degrees and then reflecting across the x axis. Find the matrix of this linear transformation. [ ] [ ] [ ] cos sin sin cos 7. linear transformation involves projecting all vectors on to the span of the vector,,. Find the matrix of this linear transformation. Let v,,. Then proj v e proj v e proj v e,,. Hence, the transformation is T x 8. Show that for an m n matrix, the set vectors {x R n x } is a subspace of R n. Let S {x R n x }. To show closure under addition, let u, v S. Then u and v. It follows that u + v u + v. Hence u + v S. To show closure under scalar multiplication, let u S and α be a scalar. Then u. It follows that αu αu α. Hence αu S. 9. Show that for an matrix, the set vectors {x R x [,, ] T } is not a subspace of R n. Let S {x R x [,, ] T }. Suppose x S. If S were a subspace, then αx must be in S for any scalar α. Setting α we see that must be an element of any subspace. Observe that [,, ] T. Hence / S. Therefore S cannot be a subspace.. Determine whether the v,,, v,, 5, v 5,, 9,and v,, span R. We must consider whether the equation c v +c v +c v +c v x has a solution for every x R. Written as an augmented matrix, this vector equation is represented by [v v v v x]. Thus the question reduces to determining whether the coefficient matrix [v v v v ] has a pivot position in every row. Observe that 5 reduces 5 9 to. Therefore, Span{v, v, v, v } R.. Show the following: x..
a If {v, v, v } is linearly independent, then so is {v, v }. b If {v, v, v } is linearly dependent, then so is {v, v, v, v }. a If {v, v, v } is linearly independent, then c v + c v + c v only has the solution c c c. Hence c v + c v c v + c v + v only has the solution c c. Thus {v, v } is linearly independent. b If {v, v, v } is linearly dependent, then there is a solution to c v + c v + c v where at least one c i for i {,, }. Hence c v + c v + c v + c v has this same solution with the addition c. Therefore {v, v, v, v } is linearly dependent.