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Math 203 Practice Final Exam

Please work out all of the following problems. Credit will be given based on the progress that you make towards the final solution. Show your work. No calculators allowed for this page.

Printable Key

Problem 1

Let

$ A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} $

Find an orthogonal matrix P and a diagonal matrix D with A = PDP^-1.

Solution

Problem 2

Use the permutation definition of the determinant to find the determinant of

$ \begin{pmatrix} 4 & 2 & 3 \\ 0 & 0 & -2 \\ 1 & 6 & 5 \end{pmatrix} $

Solution

Problem 3

Find the inverse of A if

$ A = \begin{pmatrix} 1 & 1 & 0 \\ -1 & 0 & 1 \\ 3 & -1 & 5 \end{pmatrix} $

Solution

Calculators are permitted on this part

Problem 4

Consider the matrix

$ \begin{pmatrix} 5 & 10 & 0 & 3 \\ 3 & 6 & 1 & 1 \\ 2 & 4 & 0 & 1 \\ 1 & 2 & 0 & -1 \end{pmatrix} $

A. Determine the rank of A.

Solution

B. Find a basis for the null space of A.

Solution

C. Find a basis for the column space of A using columns of A.

Solution

Problem 5 Let $ L: P_2 \rightarrow R^2 $ be defined by

$ L(f(t)) = ( f'(1), \int_0^1 f(x) dx $

A. Prove that L is a linear transformation.
Solution

B. Let S = {x² + x, x² + 1, x} and T = {(1,1), (1,2)} be bases for P₂ and R². Find the matrix for L with the bases S and T.

Solution

Problem 6

Show that the set

$ S = \{ \begin{pmatrix} 1 & 2 \\ 0 & 0 \end{pmatrix} $ , $ \begin{pmatrix} 0 & 0 \\ 2 & 1 \end{pmatrix} $ , $ \begin{pmatrix} 0 & 2 \\ 0 & 1 \end{pmatrix} $ , $ \begin{pmatrix} 2 & 0 \\ 1 & 0 \end{pmatrix} $ ,

is a basis for M^2x

².

Solution

Problem 7 Use matrices to find the unknown currents in the given circuit.

$Electric Circuit, starts at a goes through a 1 ohm resister to B, then a 4 ohm resister to C, then a 50 volt battery to d, then a 10 volt battery to d. Fron b to d it it goes through a 60 volt battery and a 2 ohm resister.$

Solution

Problem 8

Graph the equation and write the equation in standard form.

4x² + 2xy + 4y² = 15

Solution

Problem 9

One of the following is a subspace of the space of differentiable functions.

I. {f | f(0) – f ‘(0) = 1} II. {f | f(1) = f ‘(1)}

A. Determine which is not a subspace and explain why.
Solution

B. Prove that the other one is a subspace.

Solution

Problem 10

Prove that if A is a matrix such that A² = 0 then 0 is an eigenvalue for A.

Solution

Problem 11

Answer the following true or false. If it is true, explain why. If it is false explain why or provide a counter example.

A. If S = {v₁, v₂} is a linearly independent set of vectors in R³ and v₃ is not in the span of S, then {v₁, v₂, v₃} is a basis for R³.

Solution

B. Every orthonormal set of five vectors in R⁵ is a basis for R⁵.
Solution

C. Let A and B be matrices such that A²v = a, B²v = b, and ABv = c. Then

(A + B)² v = a + b + 2c
Solution

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