math 2568 - midterm 1 namemath 2568 - midterm 1 name: autumn 2014 oguz kurt problem 1 (15 pts) write...
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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
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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
![Page 3: Math 2568 - Midterm 1 NameMath 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: Problem](https://reader030.vdocuments.mx/reader030/viewer/2022040919/5e94ed285d2ae70b1b14da96/html5/thumbnails/3.jpg)
Problem 3
(15 pts) Show whether the following vectors are linearly independent or not. :
S ={x+ 3, x2 − 3, x3 − x, x3 + x2
}
![Page 4: Math 2568 - Midterm 1 NameMath 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: Problem](https://reader030.vdocuments.mx/reader030/viewer/2022040919/5e94ed285d2ae70b1b14da96/html5/thumbnails/4.jpg)
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
![Page 5: Math 2568 - Midterm 1 NameMath 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: Problem](https://reader030.vdocuments.mx/reader030/viewer/2022040919/5e94ed285d2ae70b1b14da96/html5/thumbnails/5.jpg)
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.
![Page 6: Math 2568 - Midterm 1 NameMath 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: Problem](https://reader030.vdocuments.mx/reader030/viewer/2022040919/5e94ed285d2ae70b1b14da96/html5/thumbnails/6.jpg)
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.