ENGR 115 University of California Davis Honor Code Statement Linear Algebra Worksheet

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Math 410, Linear Algebra
Dr. B Truong
Name: __________________________
Sign: ___________________________
(I work by my-self, alone and… lonely!)
EXAM 3 (Spring 2021)
Turn in 5/16/2021
This exam 3 is served as a Final Exam. There are 2 more questions will be assigned by 5/16.
This exam worths 110 points, of which 5 points are for bonus. Please present your paper in a
well organized form. If you make my life miserable to read your unorganized work, I will make
your exam miserable. Have fun.
1) (20) Define an inner product space for continuous functions on [0 , 1] as
1
 f , g  =  f ( x) g ( x)dx with the usual norm.
0
If f ( x) = 2 , g ( x) = x + 1 and h( x) = 1 − x 2
a) Find the angle θ between g and h
b) Apply the Gram-Schmidt process to find the orthonormalize set of these functions.
2) (20) For the standard Inner Product Space on M 22 , given
 3 − 2
− 1 3
A=
and B = 


4 8 
 1 1
a) Find the angle and the distance of A and B
a) Find the orthonormal basis for the matrices
3) (20) Let p( x) = a0 + a1 x + a2 x 2 + a3 x 3 and q( x) = b0 + b1 x + b2 x 2 + b3 x 3 be vectors in
P3 with the standard inner product space on P3 defined by
 p( x), q( x) = a0b0 + a1b1 + a2b2 + a3b3 .
f − g , angle of f ( x) and h( x)
b) Use the Gram-Schmidt process to form an orthogonal set for the functions
f ( x) = 1 − x − x 2 + 2 x 3 , g ( x) = −2 + 2 x − x 3 and h(x) = x − x 2 + 2 x 3
a)
Find the distance between f and g
4) (15) Given set of vectors
1 
1
1 




v1 = 0 , v 2 = 2 , v3 = 1 in R 3
2
1
0
a) Make the vectors v2 , v3 in to an orthonormalized set w2 , w3
b) Find the orthogonal projection of v1 onto the plane spanned by w2 , w3
c) Find the component of v1 orthogonal to the plane spanned by w2 , w3 .
1
Confirm that the unit vectors w1 =
v1
is orthogonal to the plane and together with
v1
w2 , w3 make it an orthonormal set w1 , w2 , w3  for v1 , v 2 , v3  .
5) (25) Given
− 5 2 3 
Q =  2 1 0 
 3 0 − 1
a) (10) Find a non-singular matrix P such that P −1Q P = D is diagonal. Find D.
b) (7) Find the spectral decomposition of Q
c) (8) Identify a quadratic form associated with this matrix.
Make an orthogonal rotation that eliminate the cross terms in this quadratic form Q.
Express Q in terms of new variables x, y, z. Identify and specify the new surface on the
new axis system for Q = 28 .
6 
1 


6) (10)
Let u = 2 and v = 2 . Show that among all the scalar multiples cv of the
4
0
vector v, the projection of u onto v is the vector closest to u, that is, show that
d (u, pro v u) is a minimum.
2
Math 410, Linear Algebra
Dr. B Truong
Name: __________________________
Sign: ___________________________
(I work by my-self, alone and… lonely!)
FINAL EXAM (Spring 2021)
Turn in 5/22/2021
This Final Exam Extra is served as a Final Exam. This exam worths 25 points
Please present your paper in a well organized form.
1)
The set of all nxn matrices A such that the linear system Ax = 0 has only trivial solutions
are the subspace of M mn . Give a counter example if it is not
T
F
2)
If S is a spanning set subspace of R n then the dimension of S must equal n .
3)
T
F
The functions f ( x) = x − 1 and g ( x) = 2 − 2 x are linearly dependent. Justify your
Answer
T
4)
5)
F

1
 1 
 1   1  
The set of vectors S = 
 − 1,
 0  forms an orthonormal basis.
 2  1  3  − 1
 
 

T
F
 1   2   1 
     
The set of vectors S =  2 ,  1 ,  3  spans a plane.
 − 2   2   − 4 
     
T
F
6) Determine whether the matrix is positive, negative definite or indefinite
7 4 7 
A = 4 7 4
4 4 7
7) Find an equivalent quadratic form by using a diagonal rotation, (by the principle axis
theorem. I dentify the conic in the new form.
Q = 2 x 2 + 4 y 2 + 6 yz − 4 z 2 = 80
1

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