Math 232 Name: Key Dr. Beezer Quiz R Fall 2005

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1 Math Name: Key Dr. eezer Quiz R Fall 00 Show all of your work and explain your answers fully. There is a total of 90 possible points. If you use a calculator on a problem be sure to write down both the input to and output from the calculator.. Let S be the vector space of symmetric matrices. uild the matrix representation of the linear transformation T : P S relative to the bases and C and then use this matrix representation to compute T ( + x x ). ( points) { + x + x + x } C T ( a + bx + cx ) [ ] a b + c a + b c a + b c a c {[ ] 0 [ ] 0 [ ]} Solution: Input to T the vectors of the basis and coordinatize the outputs relative to C [ ] [ ] [ ]) ρ C (T ()) ρ C ρ C + + [ ] [ ] [ ]) ρ C (T ( + x)) ρ C ρ C + + ρ C T + x + x )) [ ] [ ] [ ]) ρ C ρ 0 C MC T 0 To compute T ( + x x ) employ Theorem FTMR T ( + x x ) ρ C M T C ρ + x x )) ρ C M T C ρ ( )() + 7( + x) + ( )( + x + x ) )) ρ C 7 0 ρ C 0 [ ] 0 ( ) + 0 [ ] 0 0 [ ] + 0 [ ] You can of course check your answer by evaluating T ( + x x ) directly.

2 . Use a matrix representation to determine if the linear transformation T : P M surjective. ( points) T ( a + bx + cx + dx ) [ ] a + b + c + d a b + 6c d a + b c + d a + c + d Solution: Choose bases and C for the matrix representation { x x x } {[ ] [ ] C [ ] 0 [ ]} Input to T the vectors of the basis and coordinatize the outputs relative to C ρ C (T ()) ρ C ρ C ( ) ρ C (T (x)) ρ C ρ 0 C )) 6 ρ C ρ C )) ρ C ρ C [ ] 0 + [ ] 0 + ( ) [ ] [ ] 0 + ( ) MC T 6 0 [ ] + [ ] + [ ] + ( ) [ ] + [ ] + 0 [ ] [ ] + 0 [ ] + 0 [ ]) [ ]) [ ]) [ ]) 0 6 Properties of this matrix representation will translate to properties of the linear transformation The matrix representation is nonsingular since it row-reduces to the identity matrix (Theorem NSRRI) and therefore has a column space equal to C (Theorem CNSM). The column space of the matrix representation is isomorphic to the range of the linear transformation (Theorem RCSI). So the range of T has dimension equal to the dimension of the codomain M. y Theorem ROSLT T is surjective.

3 . Find a basis for the kernel of the linear transformation T : P M. T ( a + bx + cx ) [ ] a + b c a + b a + b c a + b + c Solution: Choose bases and C for the matrix representation { x x } {[ ] [ ] C [ ] 0 [ ]} Input to T the vectors of the basis and coordinatize the outputs relative to C ρ C (T ()) ρ C ρ C ρ C (T (x)) ρ C ρ C )) 0 ρ C ρ C ( ) [ ] 0 + [ ] 0 + [ ] [ ] + ( ) [ ] + [ ] + 0 [ ] + 0 [ ] + ( ) [ ]) [ ]) [ ] + 0 [ ]) 0 MC T 0 The null space of the matrix representation is isomorphic (via ρ ) to the kernel of the linear transformation (Theorem KNSI). So we compute the null space of the matrix representation by first row-reducing the matrix to Employing Theorem NS we have N ( MC T ) Sp We only need to uncoordinatize this one basis vector to get a basis for K(T ) K(T ) Sp ρ Sp ({ + x + x })

4 . The linear transformation R: M M is invertible. Use a matrix representation to determine a formula for the inverse linear transformation R : M M. ( points) R ([ a b ]) [ ] a + b a + b Solution: Choose bases and C for M and M (respectively) {[ 0 ] [ ]} {[ ] [ ]} 0 C The resulting matrix representation is [ ] MC R This matrix is invertible (its determinant is nonzero Theorem SMZD) so by Theorem IMR we can compute the matrix representation of R with a matrix inverse (Theorem TTMI) [ ] [ ] MC R To obtain a general formula for R use Theorem FTMR ([ ])) ([ ] [ ]) R x x x ρ y MC R ρ C ρ y y ([ ]) x + y ρ x y [ ] x + y x y. Let S be the vector space of symmetric matrices. Find a basis for S composed of eigenvectors of the linear transformation Q: S S. ( points) ([ ]) [ ] a b a + 8b + 0c 6a b 0c Q b c 6a b 0c a 9b c Solution: Use a single basis for both the domain and codomain since they are equal. {[ ] [ ] [ ]} The matrix representation of Q relative to is 8 0 M M Q We can analyze this matrix with the techniques of Section EE and then apply Theorem EER. The eigenvalues of this matrix are λ with eigenspaces 6 E M ( ) Sp E M () Sp E M () Sp ecause the three eigenvalues are distinct the three basis vectors from the three eigenspaces for a linearly independent set (Theorem EDELI). Theorem EER says we can uncoordinatize these eigenvectors to obtain eigenvectors of Q. y Theorem ILTLI the resulting set will remain linearly independent. Set C ρ 6 ρ ρ {[ ] 6 [ ] [ ]} Then C is a linearly independent set of size in the vector space M which has dimension as well. y Theorem G C is a basis of M.

5 6. Suppose that T : U V is a linear transformation and and C are ( bases ) for U and V (respectively). The proof of Theorem KNSI shows that the vector spaces K(T ) and N MC T are isomorphic by employing the isomorphism ρ. Show that ρ produces outputs in N MC T when supplied inputs from K(T ). That is prove that if u K(T ) then ρ (u) N MC T. ( points) Solution: Suppose that u K(T ). Then MCρ T (u) ρ C (T (u)) Theorem FTMR ρ C (0) u K(T ) 0 Theorem LTTZZ This says that ρ (u) N MC T as desired.

5.) For each of the given sets of vectors, determine whether or not the set spans R 3. Give reasons for your answers.

5.) For each of the given sets of vectors, determine whether or not the set spans R 3. Give reasons for your answers. Linear Algebra - Test File - Spring Test # For problems - consider the following system of equations. x + y - z = x + y + 4z = x + y + 6z =.) Solve the system without using your calculator..) Find the