Vector Spaces: Exercise Problems

EXERCISE PROBLEMS FOR VECTOR SPACES

1. Let V denotes the set of ordered pairs of real numbers, If (a₁, a₂) and (b₁, b₂) are elements of V and c ∈ R define

(a₁, a₂) + (b₁, b₂) = (a₁+b₁, a₂ b₂) and

c (a₁, a₂) = (ca₁, a₂)

Is V a vector space over R with these operations. Justify your answer.

[Ans: No: (x+y=0 condition is not satisfied)]

2. Let V= {(a₁, a₂); a₁, a₂ ∈ F}, where F is a field. Define addition of elements of V coordinatewise and for c ∈ F and (a₁, a₂) ∈ V, define c (a₁, a₂)= (a₁, 0). Is V a vector space over F with these operations? Justify your answer.

[Ans: No: (1.x=x for x ∈ V is not satisfied)]

3. Determine whether the following sets are subspaces of R³ under the operations of addition and scalar multiplication defined on R³. Justify your answer.

 (a) W₁ = { (a₁, a₂, a₃) ∈ R³ ; a₁ = 3ɑ₂ and a₃ =− a₂ } [Ans: Yes]

 (b) W₂ = {(a₁, a₂, α₃) ∈ R³ ; 2α₁ − 7α₂ + α₃ = 0}[Ans: Yes]

 (c) W₃ = { (a₁, a₂, a₃) ∈ R³ ; a₁ +2a₂ − 3a₃ = 1 } [Ans: No]

4. For each of the following lists of vectors in R³, determine whether the first vector can be expressed as a linear combination of the other two.

(a) (−2, 0, 3), (1, 3, 0), (2, 4, −1)        [Ans: Yes]

(b) (3, 4, 1), (1, 2, 1), (−2, 1, 1)           [Ans: No]

 (c) (5, 1, 5), (1, −2, − 3), (−2, 3,−4)           [Ans: No]

(d) (−2, 2, 2), (1, −2, −1), (−3, −3, 3)          [Ans: Yes]

5. For each list of polynomials in P₃(R), determine whether the first polynomial can be expressed as a linear combination of the other two.

 (a) x³−3x+5, x³ + 2x² −x + 1, x² + 3x² – 1      [Ans: Yes]

 (b) − 2x³− 11x² + 3x + 2, x³ − 2x² + 3x − 1, 2x³ +x²+3x−2     [Ans: Yes]

 (c) x³− 8x²+4x, x³− 2x² + 3x − 1, x³− 2x + 3.     [Ans: No]

6. In each part, determine whether the given vector is in the span of S.

(a) (2,−1,1), S={(1, 0, 2), (1, 1, 1)}         [Ans: Yes]

(b) (−1, 1, 1, 2), S= {(1, 0, 1, 1), (0, 1, 1, 1)}        [Ans: No]

(c) − x³ + 2x² + 3x + 3, S = { x³ + x² + x + 1, x² + x + 1, x + 1 }     [Ans: Yes]

 [Ans: Yes]

7. Show that the matrices  generate M_2×2 (F).

8. Show that if  then the span of {M₁, M₂, M₃} is the set of all symmetric 2×2 matrices.

9. Determine whether the following sets are linearly dependent or linearly independent.

(a)  in M_2×2 (R)

 [Ans: Linearly dependent]

(b) { x³+2x², −x²+3x+1, x³−x²+2x− 1 } in P₃ (R)

 [Ans: Linearly independent]

(c) {(1, −1, 2), (1, 2, 1), (1, 1, 4) } in R³.

[Ans: Linearly dependent]

(d)  in M_2×2 (R)

 [Ans: Linearly dependent]

(e) x²−x³+5x²−8x+6, − x²+x³− 5x² + 5x − 3,

 x²+3x² − 3x+5, 2x²+3x³ +4x² −x+1, x³−x+2

- in P4 (R).

[Ans: Linearly independent]

10. Determine which of the following sets are bases for R³.

(a) {(1, 0, −1), (2, 5, 1), (0, −4, 3)}      [Ans: Yes]

(b) {(1, 2, − 1), (1, 0, 2), (2, 1, 1) }      [Ans: Yes]

(c) {(1,−3,−2), (−3, 1, 3), (−2, −10,−2)}   [Ans: No]

11. Determine which of the following sets are bases for P₂(R).

(a) {−1−x+2x², 2+x−2x², 1−2x+4x² }      [Ans: No]

(b) {1−2x−2x², −2+3x−x², 1−x+6x²)        [Ans: Yes]

(c) { 1+2x−x², 4 − 2x + x², − 1 + 18x – 9x² }    [Ans: No]

12. Do the polynomials x³−2x²+1, 4x²−x+3 and 3x−2 generate P₃ (R)?

[Ans: No]

13. Is {(1,4, − 6), (1, 5, 8), (2, 1, 1), (0, 1, 0) } a linearly independent subset of R³? Justify your answer. [Ans: No]

14. Show that the vectors (1, 1, 0), (1, 0, 1) and (0, 1, 1) generate F³.

15. The vectors u₁ = (2,−3, 1), u₂ = (1,4,−2), u₃ = (−8, 12,−4), u₄=(1,37,−17) and u₅ = (−3, −5, 8) generate R³. Find a subset of the set {u₁, u₂, u₃, u₄, u₅) that is a basis for R³.

[Ans: { u₁, u₂, u₅}]

16. The set of solutions to the system of linear equations, x₁−2x₂+x₃=0 and 2x₁−3x₂+x₃=0 is a subspace of R³. Find a basis. [Ans: (1, 1, 1)]

17. Use the Lagrange interpolation formula to find the polynomials of smallest degree for the following points.

(a) (−2, 3), (−1,−6), (1, 0), (3,−2)       [Ans: −x³+2x²+4x−5]

(b) (−4, 24), (1, 19), (3, 3)         [Ans: −3x+12]

(c) (−1,7), (1, 5), (2, 15)        [Ans: 1/3(11x² − 3x+7)]

(d) (1,24), (3,120), (5,336), (7,720)      [Ans: x³+6x²+11x+6]