Linear Algebra (Math 220)
Assignment due Tuesday, April 22

1.  Preparation

Expect a quiz.

Relevant Reading:

Lay §§ 6.3 – 6.4

Hefferon § 3.VI

2.  Exercises

1. Let $U$ denote the $3\times 3$ matrix $$\frac{1}{3}\left(\begin{array}{ccc}\hfill 2& \hfill 2& \hfill 1\\ \hfill -2& \hfill 1& \hfill 2\\ \hfill -1& \hfill 2& \hfill -2\end{array}\right)\text{,}$$ and let $\phi$ be the linear function from ${\mathbf{R}}^3$ to ${\mathbf{R}}^3$ defined by $\phi \left(x\right)=Ux$ for all $x$ in ${\mathbf{R}}^3$.

   1. Show that the columns of $U$ are mutually perpendicular vectors in ${\mathbf{R}}^3$ of length $1$.

   2. Show that the rows of the transposed matrix ${}^{t}U$ are mutually perpendicular vectors in ${\mathbf{R}}^3$ of length $1$.

   3. Compute the matrix product ${}^{t}UU$.

   4. Show that $\phi$ is an invertible linear function, and find the matrix for ${\phi }^{-1}$.

   5. Explain why the function $\phi$ preserves lengths and angles. Hint. What effect does applying $\phi$ have on the “dot product” of two vectors?

2. Let $P_2$ be the vector space of polynomials of degree at most $2$. Define a scalar product (analogous to “dot” product) $\Gamma$ on $P_2$ with the formula $$\Gamma \left(f,g\right)={\int }_0^{1}f\left(t\right)g\left(t\right)dt\text{.}$$ Find the orthogonal complement, relative to $\Gamma$, of the subspace consisting of the constant polynomials.

3. Find the matrix, relative to the standard basis of ${\mathbf{R}}^3$, of the linear map from ${\mathbf{R}}^3$ to ${\mathbf{R}}^3$ that for each $x$ in ${\mathbf{R}}^3$ sends $x$ to its orthogonal projection on the plane in ${\mathbf{R}}^3$ defined by the linear equation $$2x-y+2z=0\text{.}$$