LINEAR ALGEBRA (Math 211)
HW #14 (Due Wed. Nov. 29 )


1. Determine which of the following transformations are ${\mathbb R}^3$ into ${\mathbb R}^3$ linear and find the corresponding matrices with respect to the standard basis for ${\mathbb R}^3$.

(a) T(x,y,z)=(x,0,z)                 (d) T(x,y,z)=(x+y,2x+z,3x+y-z)
(b) T(x,y,z)=(1,z,x-y)                 (e) T(x,y,z)=(3x+5y,3x+5y,3x+5y)
(c) T(x,y,z)=(x,y,1/z)                 (f) T(x,y,z)=(x-1,y,z+1)


2. Find the matrices of the following linear transformations from ${\mathbb R}^3$ into ${\mathbb R}^2$ with respect to the standard bases:
(a) projection along the first coordinate axis;         (b) projection along the second coordinate axis.


3. Find the matrices of the following linear transformations from ${\mathbb R}^2$ into ${\mathbb R}^2$ with respect to the standard basis for ${\mathbb R}^2$ :
(a) reflection about the line 2x-3y=0;         (b) rotation by the angle $-60^{\circ}$.


4. Consider the space P4={P(x)= ax4+bx3+cx2+dx+e} of polynomials of degree $\leq 4$. Find the matrix of the linear operator $T:P(x)\mapsto xP'(x)$ with respect to the standard basis x4, x3, x2, x, 1.


5. Let $T:{\mathbb R}^n\to {\mathbb R}^n$ be a linear operator defined by the formula $T(\mbox{\bf x})={\lambda}\mbox{\bf x}$ for a given number ${\lambda}$ and any $\mbox{\bf x}$ from ${\mathbb R}^n$. Find the matrix of T with respect to the standard basis for ${\mathbb R}^n$.