Vector spaces and linear maps – university Year 1 maths test to download as PDF

Decide whether \(U=\{(x,y,z):x=2y\}\) and \(V=\{(x,y):x\ge0\}\) are subspaces. Justify.

Show that \((1,1,1)\), \((1,2,3)\) and \((1,4,9)\) form a basis of \(\mathbb R^3\), and find the coordinates of \((3,7,13)\).

Let \(T(x,y,z)=(x+y,\,y+z)\).

1. Write the matrix of \(T\). (1 mark)
2. Find a basis of the kernel and of the image. (2 marks)
3. Verify rank-nullity. (1 mark)

A \(3\times5\) matrix \(A\) has rank 2 and represents a measurement system on \(\mathbb R^5\).

1. Find the nullity. (1 mark)
2. Find the dimension of the image and say whether the system can produce every vector of \(\mathbb R^3\). (2 marks)
3. Is the map injective? Justify. (1 mark)

In \(P_2\) with basis \(\{1,x,x^2\}\) consider \(p_1=1+x\), \(p_2=x+x^2\) and \(p_3=1+x^2\).

1. Write the coordinate vectors and show that \(p_1,p_2,p_3\) form a basis. (2 marks)
2. Express \(q=2+3x+x^2\) in terms of \(p_1,p_2,p_3\). (2 marks)