Vector spaces and linear maps – university Year 1 maths worksheet with answers to download as PDF

A subset \(W\) of \(\mathbb R^n\) is a subspace if it contains \(\mathbf 0\), and is closed under addition and scalar multiplication. Decide whether each set is a subspace of \(\mathbb R^3\) or \(\mathbb R^2\), with a proof or a counterexample.

1. \(W_1=\{(x,y,z):x+y+z=0\}\) [4 marks]
2. \(W_2=\{(x,y):xy=0\}\) [3 marks]
3. \(W_3=\{(x,y,z):x+y+z=1\}\) [2 marks]

Answer each part.

1. Is \((3,5)\) in the span of \((1,1)\) and \((1,2)\)? [3 marks]
2. Are \((1,2,3)\), \((0,1,4)\) and \((5,6,0)\) linearly independent? [3 marks]
3. Show that \((1,2,3)\), \((4,5,6)\) and \((7,8,9)\) are linearly dependent by finding a relation. [3 marks]

Let \(\mathbf b_1=(1,1,0)\), \(\mathbf b_2=(0,1,1)\) and \(\mathbf b_3=(1,0,1)\).

1. Show that \(\{\mathbf b_1,\mathbf b_2,\mathbf b_3\}\) is a basis of \(\mathbb R^3\). [3 marks]
2. Find the coordinates of \((2,3,4)\) in this basis. [4 marks]
3. What is the dimension of \(\mathbb R^3\)? How do you know it from the basis? [1 mark]

State the dimension of each vector space and justify it by giving a basis.

1. \(P_3\), the polynomials of degree at most 3. [2 marks]
2. \(M_{2\times3}\), the \(2\times3\) matrices. [2 marks]
3. \(W=\{(x,y,z):x+y+z=0\}\). [3 marks]
4. The symmetric \(2\times2\) matrices. [3 marks]

Answer each part.

1. Show that \(T(x,y)=(x+y,\,2x-y)\) is linear and write its matrix. [4 marks]
2. Find \(\det\) of this matrix and say whether \(T\) is invertible. [2 marks]
3. Show that \(S(x,y)=(x+1,y)\) is not linear. [2 marks]

Let \(A=\) \(\begin{pmatrix}1&2&3\\2&4&6\\1&1&1\end{pmatrix}\) and \(T(\mathbf x)=A\mathbf x\).

1. Find a basis for the kernel of \(T\). [4 marks]
2. Find a basis for the image of \(T\). [3 marks]
3. Verify the rank-nullity theorem. [2 marks]

A printer mixes three base inks whose colour vectors (in arbitrary units) are \(\mathbf p_1=(2,1,0)\), \(\mathbf p_2=(0,1,1)\) and \(\mathbf p_3=(1,0,1)\). A mix \(a\mathbf p_1+b\mathbf p_2+c\mathbf p_3\) uses non-negative amounts.

1. Show that the inks are linearly independent.
2. Find the amounts needed to produce the target colour \((5,4,3)\).
3. Explain why a unique recipe exists for every colour vector.
4. Which target colours need a negative amount of some ink? Give an example.

Three foods give the following amounts of (protein, carbohydrate, fat) per serving: \(\mathbf f_1=(1,2,0)\), \(\mathbf f_2=(0,1,2)\) and \(\mathbf f_3=(1,0,1)\) (in tens of grams).

1. Show that \(\{\mathbf f_1,\mathbf f_2,\mathbf f_3\}\) is a basis of \(\mathbb R^3\).
2. Find the servings that give \((3,7,7)\).
3. Describe the span of \(\mathbf f_1\) and \(\mathbf f_2\) alone: what shape is it?
4. Could the diet \((3,7,7)\) be produced from \(\mathbf f_1\) and \(\mathbf f_2\) only?

A monitoring system has six sensor channels. A \(4\times6\) matrix \(A\) transforms the channel readings \(\mathbf x\in\mathbb R^6\) into four summary values \(A\mathbf x\in\mathbb R^4\). An engineer finds that \(\operatorname{rank}A=3\).

1. What is the dimension of the image of \(A\)?
2. Use rank-nullity to find the nullity.
3. Interpret the nullity: how many independent directions of change in the readings are invisible in the summary values?
4. Can the summary values \(A\mathbf x\) cover all of \(\mathbb R^4\)? Explain.

The height of a ball is the polynomial \(h(t)=h_0+v_0t-4.9t^2\), which can be seen as a vector \((h_0,v_0,-4.9)\) in \(P_2\) with basis \(\{1,t,t^2\}\).

1. Write the matrix of the differentiation map \(D:P_2\to P_2\) with respect to this basis.
2. Find the kernel and image of \(D\) and verify rank-nullity.
3. Find \(D(h)\) for \(h=2+20t-4.9t^2\) using the matrix.
4. What physical quantity does \(D(h)\) represent? What does \(D^2h\) give?

To make italic text, a graphics program applies the shear \(S(x,y)=(x+0.5y,\,y)\) to each point of a letter.

1. Write the matrix of \(S\).
2. Find the images of the corners \((0,0)\), \((1,0)\), \((0,2)\) and \((1,2)\) of a rectangle, and describe the image.
3. Find \(\det S\) and explain what it says about the area of the letters.
4. Find \(S^{-1}\) and describe what it does.

On a chessboard-style grid a bishop moves along the diagonals. It is natural to use the basis \(\mathcal B=\{\mathbf d_1,\mathbf d_2\}\) with \(\mathbf d_1=(1,1)\) and \(\mathbf d_2=(1,-1)\).

1. Show that \(\mathcal B\) is a basis of \(\mathbb R^2\).
2. Find the coordinates of the square \((3,1)\) in this basis.
3. Find the coordinates of \((2,6)\).
4. The reflection in the line \(y=x\) swaps \(x\) and \(y\). Find its matrix in the basis \(\mathcal B\).

A lighthouse beam turns anticlockwise by \(90^\circ\) every minute. On a map with the lighthouse at the origin, the beam direction is a vector in \(\mathbb R^2\) and each minute the map \(R\) is applied to it.

1. Find the matrix of \(R\).
2. Find the kernel and the image of \(R\). Is \(R\) an isomorphism?
3. The beam points along \((3,1)\) now. Find its direction after 1, 2, 3 and 4 minutes.
4. Explain why \(R^4\) is the identity map.

Propane burns in oxygen: \(a\,\mathrm{C_3H_8}+b\,\mathrm{O_2}\to c\,\mathrm{CO_2}+d\,\mathrm{H_2O}\). Balancing the atoms gives a homogeneous linear system.

1. Show that the conditions are \(3a-c=0\), \(8a-2d=0\) and \(2b-2c-d=0\).
2. Write the system as \(A\mathbf x=\mathbf 0\) and find the kernel of \(A\).
3. What is the dimension of the kernel? Why does it make sense to expect it?
4. Write the smallest whole-number balanced equation.

Two flat walls of a model building are represented by the subspaces \(U=\{(x,y,z):x+y+z=0\}\) and \(W=\{(x,y,z):x=y\}\) of \(\mathbb R^3\).

1. Find a basis of \(U\) and a basis of \(W\).
2. Find \(U\cap W\) and its dimension.
3. Use the formula \(\dim(U+W)=\dim U+\dim W-\dim(U\cap W)\) to find \(\dim(U+W)\).
4. Show that \(U\cup W\) is not a subspace.

A shop records its sales (in hundreds) in three consecutive months: 1, 3 and 7. We model the sales by a polynomial \(p(t)=a+bt+ct^2\) with \(t=0,1,2\). Consider the map \(T:P_2\to\mathbb R^3\), \(T(p)=(p(0),p(1),p(2))\).

1. Show that \(T\) is linear and write its matrix with respect to the basis \(\{1,t,t^2\}\).
2. Show that \(T\) is an isomorphism by computing the determinant.
3. Find \(p\) with \(T(p)=(1,3,7)\).
4. Use \(p\) to predict sales in month 3. Comment on the reliability.

Four sensors send signals \(\mathbf x=(x_1,x_2,x_3,x_4)\). Three monitors compute \(A\mathbf x\) with \(A=\) \(\begin{pmatrix}1&2&0&1\\0&1&1&1\\1&3&1&2\end{pmatrix}\).

1. Show that the third row is the sum of the first two and deduce the rank of \(A\).
2. Find a basis of the kernel.
3. Find a basis of the image.
4. Verify rank-nullity. What does the kernel mean for the monitors?

A signal generator produces the functions \(f_1(t)=1\), \(f_2(t)=\cos t\) and \(f_3(t)=\sin t\). These are vectors in the space of continuous functions on \(\mathbb R\).

1. Suppose \(a+b\cos t+c\sin t=0\) for all \(t\). Substitute \(t=0,\ \frac\pi2,\ \pi\) to show that \(a=b=c=0\).
2. What does this prove about \(f_1,f_2,f_3\)?
3. Show that \(g(t)=\cos t+\sin t\) lies in the span of \(f_2,f_3\) and write its coordinates.
4. The solution set of \(y''+y=0\) is a subspace of dimension 2. Which pair from the list is a basis?

Two tanks exchange water. The amounts of dye \((x,y)\) change in one step by the linear map \(T(x,y)=(3x+y,\,x+3y)\). Use the basis \(\mathcal B=\{(1,1),(1,-1)\}\).

1. Find \(T(1,1)\) and \(T(1,-1)\) and write each as a multiple of a basis vector.
2. Write the matrix of \(T\) in the basis \(\mathcal B\).
3. Find \(T^5(2,0)\) by writing \((2,0)\) in the basis \(\mathcal B\).
4. Interpret the two multipliers.

Let \(\mathbf u,\mathbf v,\mathbf w\) be linearly independent vectors in a real vector space. Prove that \(\mathbf u+\mathbf v\), \(\mathbf v+\mathbf w\) and \(\mathbf w+\mathbf u\) are also linearly independent.

1. Suppose \(\alpha(\mathbf u+\mathbf v)+\beta(\mathbf v+\mathbf w)+\gamma(\mathbf w+\mathbf u)=\mathbf 0\). Collect terms in \(\mathbf u,\mathbf v,\mathbf w\).
2. Use the independence of \(\mathbf u,\mathbf v,\mathbf w\) to get three equations for \(\alpha,\beta,\gamma\).
3. Solve them and finish the proof.
4. Express the change of vectors by a matrix and find its determinant. Why does the proof fail over arithmetic modulo 2?