Linear Algebra Answers

3. The geometrical representation of a vector v =



a

b

is an arrow starting at the origin and ending at the point (a, b).

Multiplication of a vector v with the matrix A =



cos θ − sin θ

sin θ cos θ



yields a vector p =Av

which is a counter clockwise rotation of v by an angle of θ.

a) Find the vector p that is obtained if v =



3

−4



is rotated counter clockwise by 40◦

.

b) Find the vector q that is obtained if v =



2

3



is rotated clockwise by 40◦

.

Determine whether W = {(x, y,z) | x + y + z + 1 = 0, x, y,z ∈ R} a subspace of R³ or

not?

The first four Hermite polynomials are f(x) = 1,g(x) = 2t, h(x) = 2−4t +t², and

p(x) =6−18t+9t² −t³. Show that these polynomials form a basis for P3.

Let x, y and z be three vectors in a vector space V

a. Give a definition of span{x,y, z} in set notation. [2]

b. Prove that the span{x,y, z} is a subspace of V.