Search & Filtering
Find all values of lambda such that ,the following system has a none-zero solutions.
2x+y= lambda x
4x-y =lambda y
Determine whether the vector (9,0,9) is a linear combination of the vectors
(1,1,0), (1,0,1), (2,1,1) and (0,1, −1).
Let W = {(x, y, z): y² = x + z}, Is W a subspace of R³
A. Let L: R3 R3 be defined by L([u₁ u₂ u3]) = [u₁ + 1, 2u₂]. Is L a linear transformation?
B. Let L: R₂ → R₂ be defined by L([u₁ u₂]) = [2u1 2u₂]. Is L a linear transformation?
Find a polynomial, P(x), of degree 3 with zeros of 4,1 and −1, if P(0) = 8
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◦
.
Find all the values of k so that the set{ ( 1,−3,2),(−3,9,−6),(5,−7,k)} form the basis for R³
For what values of α are vectors (1,1,2,1), (2,1,2,3), (1,4,2,1) (-1,3,5,α) are linearly inde-
pendent
Let S = {u1,u2,u3} be a basis for the vector space V. Show that T = {w1,w2,w3} is also
a basis for V, where w1 = u1 +u2 +u3, w2 = u2 +u3,w3 = u3.
Letv1 = (0,3,6,0),v2 = (0,2,4,6), and v3 = (1,−1,−2,1). Express (4,−1,−2,−11) as a
linear combination of v1,v2, and v3
