• 1.

  draft
  $A,\;B$ $2 \times 2$ matrices. Find eigenvalues and eigenvectors of both. Hence or otherwise, find $B^n$ for largish $n$.
• 2.

  draft
  Given a 3 x 3 matrix, and two eigenvectors find their corresponding eigenvalues. Also fnd the characteristic polynomial and using this find the third eigenvalue and a normalised eigenvector $(x=1)$.
• 3.

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

  draft
  $A$ a $3 \times 3$ matrix. Using row operations on the augmented matrix $\left(A | I_3\right)$ reduce to $\left(I_3 | A^{-1}\right)$.
• 5.

  Ready to use
  Elementary Exercises in multiplying matrices. 
• 6.

  Ready to use
  Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix. 
• 7.

  Ready to use
  Real numbers $a,\;b,\;c$ and $d$ are such that $a+b+c+d=1$ and for the given vectors $\textbf{v}_1,\;\textbf{v}_2,\;\textbf{v}_3,\;\textbf{v}_4$ $a\textbf{v}_1+b\textbf{v}_2+c\textbf{v}_3+d\textbf{v}_4=\textbf{0}$. Find $a,\;b,\;c,\;d$.
• 8.

  draft
  Given $5$ vectors in $\mathbb{R^4}$ determine if a spanning set for $\mathbb{R^4}$ or not by looking for any simple dependencies between the vectors.
• 9.

  draft
  Given the following three vectors $\textbf{v}_1,\;\textbf{v}_2,\;\textbf{v}_3$ Find out whether they are a linearly independent set are not. Also if linearly dependent find the relationship $\textbf{v}_{r}=p\textbf{v}_{s}+q\textbf{v}_{t}$ for suitable $r,\;s,\;t$ and integers $p,\;q$.
• 10.

  Doesn't work
  Given $6$ vectors in $\mathbb{R^4}$ and given that they span $\mathbb{R^4}$ find a  basis.
• 11.

  draft
  Choose which of 5 matrices are in a) row echelon form but not reduced b) reduced row echelon form c) neither.
• 12.

  Doesn't work
  Given a matrix in row reduced form use this to find bases for the null, column and row spaces of the matrix.
• 13.

  Doesn't work
  Reduce a 5x6 matrix to row reduced form and using this find rank and nullity.
• 14.

  Ready to use
  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 $\phi$.
• 15.

  Ready to use
  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 $\phi$.

There are no questions in this group.