{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# practice1 - FINAL EXAM PRACTICE I May 8 2010 Name Math 21b...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Problem 1) (20 points) True or False? No justifications are needed. 1) T F If A is a real n × n matrix, then A + A T is diagonalizable. 2) T F The equation z 3 = 1 has three different complex solutions z and z = 1 is one of them. 3) T F There is a 2 × 2 projection matrix A that projects onto a line and a 2 × 2 reflection matrix B that rotates about a point, so that AB is a rotation matrix. 4) T F The transformation T ( f )( x ) = f ( f ( x )) is linear on the space C of all smooth functions 5) T F If a continuous function f defined on the interval [ π, π ] is both even and odd, then it must be constant function. 6) T F If A is a 2 × 2 matrix, then the characteristic polynomial of AA T is λ 2 + for some real constant c . 7) T F The function f ( t ) = e t is an eigenfunction with eigenvalue 1 of the linear operator T = D 2 where Df = f . 8) T F The initial value problem f ′′ ( x ) + f ( x ) = sin( x ) , f ′′ (0) = 1 , f (0) = 1 has exactly one solution. 9) T F The transformation L ( A ) = A + A T is a linear transformation on M n and L has only the eigenvalues 0 and 2. 10) T F The set X of smooth functions f ( x, y, z ) which satisfy the f xx + f yy + f zz = f is a linear space. 11) T F If A is a 5 × 5 matrix that has rank 2, then the eigenvalue 0 has algebraic multiplicity 3. 12) T F Every equilibrium point of a nonlinear system ˙ x = f ( x, y ) , ˙ y = g ( x, y ) is located on at least one nullcline. 13) T F The transformation T ( A ) = rank( A ) is linear from the space M 2 of all real 2 × 2 matrices to R .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 8

practice1 - FINAL EXAM PRACTICE I May 8 2010 Name Math 21b...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online