Math 2568 - Midterm 1 Name:Autumn 2014 Oguz Kurt

Problem 1

(15 pts) Write the definitions (not an equivalent statement or a method!) of the following:

(a) A set of vectors v1,v2,v3,v4 are linearly dependent if

(b) A linear combination of v1,v2,v3 is

(c) The dimension of a subspace W of a vector space V is

Problem 2

(15 pts) Find the solution set for the following system of linear equations:

w + x + 2z = 12w + x + y + 3z = 43w + 2x + y + z = 1

Problem 3

(15 pts) Show whether the following vectors are linearly independent or not. :

S ={x+ 3, x2 − 3, x3 − x, x3 + x2

}

Problem 4

(20 pts) For the following problems, show whether W is a subspace of the vector space V.

(a) V = M2×2, W =

{[a bc d

]: a+ πd− b− 2c = 0

}

(b) V = R3, W =

xyz

: x− 2y +√z = 0

Problem 5

(20 pts) Find a basis for the subspace W = Span{x− 1, x2 − 1, x3 + x− 1, x2 + x, x3 + 1

}of the space of

polynomials and calculate the dimension of W.

Problem 6

(15 pts) Describe W =

{[a 0c d

]: a+ πc− d = 0

}as a span of vectors. You do not need to show whether

the spanning vectors are linearly independent or not.