Linear Algebra and Probability 2

• Consider the matrix
• Find A−1 .
• Solve the system

• Write the matrix

  ⎡ ⎤ −10 ⎣ 3 ⎦ 14

as a linear combination of the matrices

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

• Is the polynomial

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.

• Is the polynomial

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.

• Are the vectors p1, p2, p3 linearly independent?
• Does the set {p1, p2, p3} span V ?
• 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:

• Is F a linear mapping? Justify your answer.
• Is F a one-to-one mapping? Justify your answer.
• Is F an onto mapping? Justify your answer.
• Does F have an inverse? If yes, find the inverse. If not, explain why not.