John Steele

Using the reduction formulas for products of powers or sin and cosine

Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$  with coefficients in the real numbers. Let the linear map $\phi: P_4 \rightarrow P_4$ be defined by: \[\phi(p(x))=p(a)+p(bx+c).\]Using the standard basis for range and domain find the matrix given by $\phi$.

Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$  with coefficients in the real numbers.

Let the linear map $\phi: P_4 \rightarrow P_4$ be defined by:

$\phi(p(x))=ap(x) + (bx + c)p'(x) + (x ^ 2 + dx + f)p''(x)$

Using the standard basis for range and domain find the matrix given by $\phi$.

Given a matrix in row reduced form use this to find bases for the null, column and row spaces of the matrix.

Reduce a 5x6 matrix to row reduced form and using this find rank and nullity.

Find the determinant and inverse of three $2 \times 2$ invertible matrices.

Multiplication of $2 \times 2$ matrices.

Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.

Elementary Exercises in multiplying matrices. 

Three examples of determinant of 2x2 matrices.

Multiplication of two matrices.

This question tests students knowledge of basic matrix arithmetic.

Given $6$ vectors in $\mathbb{R^4}$ and given that they span $\mathbb{R^4}$ find a  basis.

Calculation of the length and alternative form of the parameteric representation of a curve.

A question to test integration by parts in a "pre-Fourier series" setting.