Paper 1
a) Let \(\mathbf u=(2,-1,3)\) and \(\mathbf v=(-1,4,2)\). Compute \(2\mathbf u-\mathbf v\).
[3 marks]b) Use a scalar product to decide whether \(\mathbf u\) and \(\mathbf v\) from part (a) are orthogonal.
[3 marks]c) Compute \(\mathbf u\times\mathbf v\) for the vectors from part (a), and verify that the result is perpendicular to both \(\mathbf u\) and \(\mathbf v\).
[5 marks]d) Use your vector product from part (c) as a normal vector to find an equation of the plane through \(P=(1,0,-2)\).
[4 marks]a) Let \(A=\begin{pmatrix}1&0&-1\\2&1&3\end{pmatrix}\) and \(B=\begin{pmatrix}2&3\\1&-2\\0&4\end{pmatrix}\). Compute \(AB\).
[4 marks]b) State the transpose rule for a product and use part (a) to find \((AB)^T\).
[3 marks]c) Use an inverse matrix to solve \(\begin{pmatrix}3&1\\2&1\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}4\\3\end{pmatrix}\).
[5 marks]d) Give a concrete example showing that matrix multiplication is not commutative.
[3 marks]a) Write the augmented matrix for the system \(x+y+z=3\), \(2x+3y+5z=8\), \(x+2y+4z=6\).
[3 marks]b) Row-reduce the augmented matrix from part (a) enough to classify the system.
[5 marks]c) Replace the third equation by \(x+2y+4z=5\). Find the solution set of the new system.
[5 marks]d) Explain the difference between a pivot variable and a free variable.
[2 marks]a) Compute \(\det A\) for \(A=\begin{pmatrix}2&1&0\\0&3&1\\1&2&4\end{pmatrix}\).
[4 marks]b) Use your determinant from part (a) to decide whether \(A\) is invertible and whether its columns are linearly independent.
[3 marks]c) Let \(B\) be obtained from \(A\) by swapping two rows, and let \(C\) be obtained from \(A\) by multiplying one row by \(5\). Find \(\det B\) and \(\det C\).
[4 marks]d) Find the area of the parallelogram spanned by \(\mathbf p=(3,-1)\) and \(\mathbf q=(1,4)\).
[4 marks]a) Show that \(W=\{(x,y,z)\in\mathbb R^3:x-2y+z=0\}\) is a subspace of \(\mathbb R^3\).
[5 marks]b) Find a basis for \(W\) and state \(\dim W\).
[5 marks]c) Decide whether \((3,2,-1)\) lies in the span of your basis from part (b).
[3 marks]d) Explain why \(S=\{(x,y,z):x-2y+z=1\}\) is not a subspace of \(\mathbb R^3\).
[2 marks]a) Let \(T:\mathbb R^2\to\mathbb R^2\) be defined by \(T(x,y)=(2x-y,x+3y)\). Find the standard matrix of \(T\).
[3 marks]b) Find \(\ker T\), \(\operatorname{im}T\), the rank, and the nullity.
[5 marks]c) Find the eigenvalues of \(M=\begin{pmatrix}4&1\\0&2\end{pmatrix}\) and one eigenvector for each eigenvalue.
[5 marks]d) Decide whether \(M\) from part (c) is diagonalizable, and give suitable matrices \(P\) and \(D\) if it is.
[2 marks]