Math 63 Linear Algebra Sample Exam Questions

Math 63 Linear Algebra Sample Exam Questions
1. Please complete the following definitions:
(a) The set of vectors {v 1 , v 2 , . . . , v n } is linearly independent if . . .
(b) The set of vectors {v 1 , v 2 , . . . , v n } is linearly dependent if . . .
(c) The mapping T : V → W is a linear transformation from the vector space V to the
vector space W if . . .
(d) A basis for the vector space V is . . .
(e) The subset H of the vector space V is a subspace of V if . . .
(f) If A is a matrix, then the null space of A is . . .
(g) If A is a matrix, then the column space of A is . . .
(h) If A is a matrix, then AT is
2. Solve the following system of equations using row reduction:
3x + 5y − 4z =
7
−3x − 2y + 4z = −1
6x + y − 8z = −4
Describe the solution set in geometric terms:


1 1 3
1


4 
3. Find bases for all the fundamental subspaces associated with A =  2 1 5
1 2 4 −1
Basis for Col A:
Basis for Row A:
Basis for Nul A:
Basis for Nul AT :






8
−4
2






4. Let v 1 =  3  and v 2 =  −5 . Is the vector v 3 =  2  in Span{v 1 , v 2 }?
−9
8
−5
5. Find the eigenvalues and bases for the corresponding eigenspaces for the matrix
A=
"
1 2
3 4
#
6. Let C 1 [0, 1] represent the vector space of continuously differentiable functions defined on the
interval [0, 1] and let C[0, 1] be the vector space of continuous functions on [0, 1]. Define the
transformation T : C 1 [0, 1] → C[0, 1] by
f → Tf
Thus, T (ex ) =
d(ex )
dx
where
(T f )(x) =
T
df
(x) − 2f (x) for 0 ≤ x ≤ 1.
dx
− 2ex = −ex , i.e, ex → −ex .
1
(a) What is the zero vector in the space C[0, 1]?
(b) Show that T is a linear transformation.
(c) Determine the kernel of T . What is dim ker T ?
7. Let B = {b1 , b2 } and C = {c1 , c2 } be two bases for R2 where
b1 =
"
2
1
#
,
b2 =
"
#
1
2
,
c1 =
"
1
1
#
,
c2 =
"
1
−1
#
.
Determine the matrix for the linear transformation that maps a vector x in standard coordinates to its C-coordinates [x]C :


PC−1 = 

Now determine the matrix that maps the B-coordinates of x to the C-coordinates of x:


P

C←B=

8. Let T : V → W be a linear transformation from the vector spaces V to the vector space
W . Let S = {v 1 , v 2 , . . . , v n } be a set of vectors in V . Suppose that the set of vectors
{T (v 1 ), T (v 2 ), . . . , T (v n )} is a linearly independent set of vectors in W . Prove that S must
be a linearly independent set in V . Produce a counter-example to show that the converse is
generally false.
9. Show that the set of vectors B = {1, 1 − t, 1 + t + t2 } is a basis for the vector space P 2 .
Let [x]S represent the coordinates of x ∈ P 2 relative to the standard basis S = {1, t, t2 }.
Thus, if x = 2 + 3t − 7t2 , then


2


[x]S =  3 
−7
Determine the change-of-basis transformation that will determine the B-coordinates of a
vector x from its standard coordinates. What are the B-coordinates of x = 2 + 3t − 7t2 ?
10. Let A =
"
#
4
5
. Compute A1,000,001 .
−3 −4
11. Show that the matrix A =
"
5 −5
1
1
#
is similar to the matrix
"
#
3 1
.
−1 3
12. Find the eigenvalues and bases for the corresponding eigenspaces for the matrix


0 2 2


A= 2 0 2 
2 2 0
"
#
2 2
13. Let A =
. Describe the action of the linear transformation x → Ax in precise
−2 2
geometric terms.
2
14. Prove that, if λ1 and λ2 are distinct eigenvalues of a symmetric matrix A, then the corresponding eigenspaces are orthogonal.
15. Let A be a matrix. Verify that the orthogonal complement of the column space of A is the
null space of the transpose of A, which can be concisely written as
Null AT = (Col A)⊥ .
16. Let W be the subspace of R3 spanned by the two vectors



5


v1 =  2  ,
−1
and let

1


v 2 =  −2  .
1


2


y =  1 .
3
ˆ + z where y ∈ W and z ∈ W ⊥ .
Find a decomposition y = y
17. Consider the data: (1, −1), (2, 1), (4, 2), (5, 3).
(a) Fill in the blanks to set up the least-squares problem to fit the data with the model
y = β0 + β1 x:











|


 


 


 |

{z




=


} 











{z
β
}
X

|
{z
y
}
(b) Determine the normal equations for the least-squares solution:
(c) Solve the normal equations to find the equation of the least-squares line, and fill in the
blanks (you may use your calculator to do the necessary direct matrix arithmetic for this
problem. You cannot use an automated least-squares function to detemine the answer
directly):
y=
+
x


3 −2 4


6 2 .
18. Find an orthogonal diagonalization for the matrix1 A =  −2
4
2 3
19. Classify the quadratic form Q(x) = 2x2 + 10xy + 2y 2 and make a change of variable that
transforms the quadratic form to one with no cross product term.
20. Let U be an n × n orthogonal matrix and let A be an n × n symmetric matrix. Prove the
following results:
(a) ||U x|| = ||x|| for all x ∈ Rn .
1
Helpful information: the eigenvalues of A are λ = 7, −2
3
(b) U x · U y = x · y for all x, y ∈ Rn .
(c) Ax · y = x · Ay.
21. A distribution network for electric power has 5000 nodes; each node is a transformer that
modulates the local distribution of power to a group of users. On any given day a node is
either working (up) or not working (down). If a transformer is up on a given day, the chance
that it will be down the following day is 0.001. If a transformer is down on a given day, the
chance that it will be up the next day is 0.25. On January 1, 2001 all 5,000 of these nodes
are up. Approximately how many will be working on January 1, 2004? You may use your
calculator for this problem, if you wish.
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