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practice_exam1

# practice_exam1 - (1 Suppose that the system of equations...

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MATH 3000 EXAM 1 PRACTICE SHEET Computation (1) Write the general solution to the following system of equations in parametric vector notation (for example ~x = ~a + t ~ b + s~ c , for some vectors ~a, ~ b,~ c ): x 2 + 2 x 3 = 1 x 1 + 4 x 2 + 8 x 2 = 6 3 x 1 + 2 x 2 + 4 x 3 = 8 (2) Give an example of a matrix in echelon form which is not in reduced echelon form. (3) Find a parametric equation for the plane defined by the equation a · x = 12 where a = (2 , 8 , 4). (4) Write down a system of equations in the form of an augmented matrix for finding the coefficients a, b, c in the equation for a parabola y = ax 2 + bx + c which passes through the points (1 , 1) , (2 , 3) , (5 , 7). (5) Consider the parametric description of a plane P : P = ~x 3 ~x = t 3 1 - 4 + s 1 2 1 , for some s, t Find a matrix A such that we have P = { ~x 3 | ~x = A~ y, for some ~ y 2 } . (6) Let A = 1 2 - 1 0 1 4 Give a parametric description for the solutions to the equation Ax = 0. (7) Let A = 1 2 0 2 Find A 2 . Describe the set of vectors ~x which solve the equation A

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Unformatted text preview: (1) Suppose that the system of equations described by the augmented matrix ±± a b c d ² ³ ³ ³ ³ ± p q ²² is inconsistent (has no solutions). Show that the vectors ( a,b ) and ( c,d ) must be parallel. (2) Show that ( x 1 ,x 2 ,x 3 ) ∈ Span ( (2 , 1 , 7) , (3 , 2 , 6) ) if and only if 8 x 1-9 x 2-x 3 = 0. (3) Prove that the function T ( x 1 ,x 2 ) = ( x 1-2 x 2 , 4 x 2 ) is a linear transformation (4) (a) Suppose that A is an m × n matrix with Ax = 0 for all x ∈ R m . Show that A = 0. (b) Suppose that A,B are m × n matrices with Ax = Bx for all x ∈ R m . Show that A = B . (5) Let ~x,~ y be in R n . Show that proj ~ y ~x and ~x-proj ~ y ~x are perpendicular. (6) Let ~x ∈ R n . Show that the function f : R n → R 1 = R deﬁned by f ( ~ y ) = ~x · ~ y is a linear transformation....
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