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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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