Linear Algebra and Probability 2

Linear Algebra and Probability

Question 1. Consider the matrix

A =

  3 1 −42 −3 8

7 1 3

 

(a) Find A−1. (b) Solve the system

3x + y − 4z = −10 2x− 3y + 8z = 3 7x + y + 3z = 14

(c) Write the matrix   −103

14

 

as a linear combination of the matrices  32

7

  ,

  1−3

1

  , and

  −48

3

 

Question 2. Let V be the vector space of polynomials with degree ≤ 3, and

p1 = t3 + 2t2 + 3t + 9, p2 = 2t3 + t2 − 5t + 6, p3 = 3t3 − 4t2 − 7t − 3.

(a) Is the polynomial p = 8t3 − 15t2 − 7t − 12

a linear combination of p1, p2, and p3? If yes, write p as a linear combination of p1, p2, and p3. If not, explain why not. (b) Is the polynomial

p = 8t3 − 15t2 − 7t − 6

a linear combination of p1, p2, and p3? If yes, write p as a linear combination of p1, p2, and p3. If not, explain why not. (c) Are the vectors p1, p2, p3 linearly independent? (d) Does the set {p1, p2, p3} span V ? (e) Is the set {p1, p2, p3} a basis of V ?

Question 3. Let V be the set of 2×1 matrices, and define a mapping F : V → V by matrix multiplication:[ x y

] 7→ [

3 −1 1 3

][ x y

] (a) Is F a linear mapping? Justify your answer. (b) Is F a one-to-one mapping? Justify your answer. (c) Is F an onto mapping? Justify your answer. (d) Does F have an inverse? If yes, find the inverse. If not, explain why not.

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