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MAT340 Exam 2: Linear Algebra Problems, Exams of Linear Algebra

The solutions to exam 2 for a linear algebra course, mat340. It includes problems on expressing vectors in different bases, finding matrix representations of linear transformations, and calculating the row space, column space, and null space of matrices. Students are expected to have a solid understanding of linear algebra concepts.

Typology: Exams

Pre 2010

Uploaded on 08/09/2009

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MAT340 Exam 2 Due 14 November, 2006
Prof. Thistleton
You may consult written sources such as textbooks or your class notes as you work through this
exam. You may not, however, work with or receive help from anyone on this exam.
1. Let
V={v1, v2, v3}where v1=
1
1
1
,v2=
1
1
1
,v3=
1
1
1
and
U={u1, u2, u3}where u1=
1
1
1
,u2=
1
1
0
,u3=
0
1
1
.
(a) Express x=
4
6
2
in terms of basis V.
(b) Express x=
4
6
2
in terms of basis U.
pf3
pf4
pf5

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MAT340 Exam 2 Due 14 November, 2006 Prof. Thistleton

You may consult written sources such as textbooks or your class notes as you work through this exam. You may not, however, work with or receive help from anyone on this exam.

  1. Let

V = {v 1 , v 2 , v 3 } where v 1 =

 

 , v 2 =

 

 , v 3 =

 

 

and

U = {u 1 , u 2 , u 3 } where u 1 =

 

 , u 2 =

 

 , u 3 =

 

 .

(a) Express x =

 

  in terms of basis V.

(b) Express x =

  

   in terms of basis^ U^.

(c) Find the matrix S which transforms vectors in terms of basis V to basis U and demon- strate that this works for your vector x.

  1. Define the linear operator L : <^3 → <^3 as

L(x) = L([x 1 , x 2 , x 3 ]T^ ) = [2x 1 − x 3 , x 2 + x 3 , x 1 − x 2 ]T

Find the matrix representation of this linear transformation with respect to the standard basis and use it to calculate L([1, 1 , 1]T^ ).

  1. If L : <^2 → <^2 is such that

L([3, −2]T^ ) = [2, 2]T^ and L([2, 1]T^ ) = [3, 5]T

then calculate L([10, −9]T^ ).

(d) Show that B = S−^1 AS.

  1. Let A =

  

  

Find a basis for the row space, the column space and the null space of A.

  1. Show that the vectors u = ex^ and v = x^2 are independent in the space C(−∞, ∞).
  2. Are the vectors u = x + 1, v = x − 1 and w = x^2 + x − 1 independent in the space P 3?