MATH 410 UCD Linear Algebra & Equivalent Quadratic Form Question

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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
 
 

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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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Tags:
linear algebra

Trivial solutions

linear system Ax

set of all nxn matrices

forms an orthonormal basis

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