# Math 315

## Linear Algebra and Differential Equations

## Homework Hints

### Assignment 3

**3.3.4**
Set up a system of three equations with two unknowns (alpha and beta).
Is there a solution to these equations?

**3.3.8**
You need to check that

(a) T(x+y) = T(x) + T(y) when x and y are vectors in R^2; and

(b) T(cx) = cT(x) when c is a scalar and x is a vector in R^2.

**3.4.2**
Put the coefficient matrix into reduced row echelon form.

**3.5.10**
Multiply out AB and BA in terms of a and b.
By setting the corresponding elements to be equal to one another,
you have four linear equations in four unknowns.
Find a solution.

**3.6.4**
For part (b), first do the multiplication without substituting for I and J.
You can then replace J^2 with -I and simplify.
Finally, you can substitute in for I and J and simplify.

**3.6.10**
You can write out matrices A = [a b; b c] and B = [d e;e f ].
Carry out the multiplication.
The off diagaonal elements should not be equal.
There are many choices.

Last modified: February 1, 2000

Bret Larget,
larget@mathcs.duq.edu