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2 hours 30 minutes90 marks

Level 1 - Linear Algebra Paper 1

A full Level 1 - Linear Algebra exam covering vector geometry, matrix algebra, systems, determinants, subspaces, linear maps, and eigenvalue methods.

Instructions

  • Attempt all questions.
  • All main questions carry 15 marks.
  • Show sufficient working to justify each answer.
  • Exact answers are preferred unless a decimal approximation is requested.

Section A: Vectors In Rn

1. Vector geometry and planes

15 marks
This question concerns vector operations, scalar products, vector products, and planes in \(\mathbb R^3\).
(a)
3 marks
Let \(\mathbf u=(2,-1,3)\) and \(\mathbf v=(-1,4,2)\). Compute \(2\mathbf u-\mathbf v\).
(b)
3 marks
Use a scalar product to decide whether \(\mathbf u\) and \(\mathbf v\) from part (a) are orthogonal.
(c)
5 marks
Compute \(\mathbf u\times\mathbf v\) for the vectors from part (a), and verify that the result is perpendicular to both \(\mathbf u\) and \(\mathbf v\).
(d)
4 marks
Use your vector product from part (c) as a normal vector to find an equation of the plane through \(P=(1,0,-2)\).

Section B: Matrix Algebra

2. Matrix products and inverses

15 marks
This question concerns matrix multiplication, transpose rules, inverse matrices, and non-commutativity.
(a)
4 marks
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 \(AB\).
(b)
3 marks
State the transpose rule for a product and use part (a) to find \((AB)^T\).
(c)
5 marks
Use an inverse matrix to solve \(\begin{pmatrix}3&1\\2&1\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}4\\3\end{pmatrix}\).
(d)
3 marks
Give a concrete example showing that matrix multiplication is not commutative.

Section C: Linear Systems

3. Solving and classifying systems

15 marks
This question uses augmented matrices, row reduction, and solution-set interpretation.
(a)
3 marks
Write the augmented matrix for the system \(x+y+z=3\), \(2x+3y+5z=8\), \(x+2y+4z=6\).
(b)
5 marks
Row-reduce the augmented matrix from part (a) enough to classify the system.
(c)
5 marks
Replace the third equation by \(x+2y+4z=5\). Find the solution set of the new system.
(d)
2 marks
Explain the difference between a pivot variable and a free variable.

Section D: Determinants

4. Determinants and their consequences

15 marks
This question concerns determinant calculation, determinant properties, geometry, and invertibility.
(a)
4 marks
Compute \(\det A\) for \(A=\begin{pmatrix}2&1&0\\0&3&1\\1&2&4\end{pmatrix}\).
(b)
3 marks
Use your determinant from part (a) to decide whether \(A\) is invertible and whether its columns are linearly independent.
(c)
4 marks
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 \(\det C\).
(d)
4 marks
Find the area of the parallelogram spanned by \(\mathbf p=(3,-1)\) and \(\mathbf q=(1,4)\).

Section E: Subspaces and Dimension

5. Subspaces, bases, and coordinates

15 marks
This question concerns subspaces, spans, independence, bases, and coordinates.
(a)
5 marks
Show that \(W=\{(x,y,z)\in\mathbb R^3:x-2y+z=0\}\) is a subspace of \(\mathbb R^3\).
(b)
5 marks
Find a basis for \(W\) and state \(\dim W\).
(c)
3 marks
Decide whether \((3,2,-1)\) lies in the span of your basis from part (b).
(d)
2 marks
Explain why \(S=\{(x,y,z):x-2y+z=1\}\) is not a subspace of \(\mathbb R^3\).

Section F: Linear Maps and Eigenvalues

6. Linear maps and eigenvalue structure

15 marks
This question concerns matrix representations, kernels, images, rank-nullity, and diagonalisation.
(a)
3 marks
Let \(T:\mathbb R^2\to\mathbb R^2\) be defined by \(T(x,y)=(2x-y,x+3y)\). Find the standard matrix of \(T\).
(b)
5 marks
Find \(\ker T\), \(\operatorname{im}T\), the rank, and the nullity.
(c)
5 marks
Find the eigenvalues of \(M=\begin{pmatrix}4&1\\0&2\end{pmatrix}\) and one eigenvector for each eigenvalue.
(d)
2 marks
Decide whether \(M\) from part (c) is diagonalizable, and give suitable matrices \(P\) and \(D\) if it is.