DP Further Mathematics HL Questionbank
Topic 1 - Linear Algebra
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Description
The aim of this section is to introduce students to the principles of matrices, vector spaces and linear algebra, including eigenvalues and geometrical interpretations.
Directly related questions
- 18M.1.hl.TZ0.8c: Find all the solutions to the equations when \(a\) = 3, \(b\) = 10 in the form r = s +...
- 18M.1.hl.TZ0.8b: Explain why the equations have no unique solution when \(a\) = 3.
- 18M.1.hl.TZ0.4d.ii: Give a geometric description of the transformation T.
- 18M.1.hl.TZ0.4d.i: Find the 2 × 2 matrix representing T.
- 18M.1.hl.TZ0.4c: Write down the 2 × 2 matrices representing T3, T4 and T5.
- 18M.1.hl.TZ0.2b: Let B = \(\left[ {\begin{array}{*{20}{c}} 4&2 \\ 1&{ - 3} \end{array}} \right]\). Given...
- 18M.1.hl.TZ0.2a: Show that A4 = 12A + 5I.
- 18M.1.hl.TZ0.13d: State two eigenvectors of M which correspond to the two eigenvalues.
- 18M.1.hl.TZ0.13c: State the two eigenvalues of M.
- 18M.1.hl.TZ0.13b: Explain what happens to points on the line \(4y + x = 0\) when they are transformed by M.
- 18M.1.hl.TZ0.13a: Show that the linear transformation represented by M transforms any point on the line \(y = x\)...
- 18M.1.hl.TZ0.10c: A matrix M is said to be orthogonal if M TM = I where I is the identity. Show that Q is orthogonal.
- 18M.1.hl.TZ0.10b: Describe the transformation represented by the matrix PQ.
- 18M.1.hl.TZ0.10a.ii: determine the 2 × 2 matrix Q which represents an anticlockwise rotation of θ about the origin.
- 18M.1.hl.TZ0.10a.i: determine the 2 × 2 matrix P which represents a reflection in the...
- 16M.2.hl.TZ0.8b: (i) Write down the matrices M\(_1\), M\(_2\) representing the permutations...
- 16M.1.hl.TZ0.14b: (i) Find the value of \(a\). (ii) Find the eigenvalues of N. (iii) Find...
- 16M.1.hl.TZ0.12b: (i) Show that the set of position vectors of points whose coordinates satisfy...
- 16M.1.hl.TZ0.12a: Explain why the set of position vectors of points whose coordinates satisfy \(x - y - z =...
- 16M.1.hl.TZ0.14a: (i) Explain why M is a square matrix. (ii) Find the set of possible values of det(M).
- 17M.2.hl.TZ0.9a: The point \((x,{\text{ }}y)\) is rotated through an anticlockwise angle \(\alpha \) about the...
- 17M.2.hl.TZ0.4c: Prove, using mathematical induction, that B\(^n = {8^{n - 2}}\)B\(^2\) for...
- 17M.2.hl.TZ0.4b.iii: Explain briefly how your results verify the rank-nullity theorem.
- 17M.2.hl.TZ0.4b.ii: Determine the null space of B.
- 17M.2.hl.TZ0.4b.i: Explain how it can be seen immediately that B is singular without calculating its determinant.
- 17M.2.hl.TZ0.4a.ii: Hence show that A is singular when \(\lambda = 1\) and find the other value of \(\lambda \) for...
- 17M.2.hl.TZ0.4a.i: Find an expression for det(A) in terms of \(\lambda \), simplifying your answer.
- 17M.1.hl.TZ0.3c: State the rank of the matrix of coefficients, justifying your answer.
- 17M.1.hl.TZ0.3b: For these values of \(\lambda \) and \(\mu \), solve the equations.
- 17M.1.hl.TZ0.3a: Determine the value of \(\lambda \) and the value of \(\mu \) for which the equations are...
- 17M.1.hl.TZ0.15b.ii: Express the vector \[\left[ {\begin{array}{*{20}{c}} 2 \\ 8 \\ 0 \end{array}} \right]\] as a...
- 17M.1.hl.TZ0.15b.i: Show that the...
- 17M.1.hl.TZ0.15a.ii: Explain briefly why v\(_1\), v\(_2\), v\(_3\) form a basis for vectors in \({\mathbb{R}^3}\).
- 17M.1.hl.TZ0.15a.i: By considering \({\alpha _1}\)v\(_1 + {\alpha _2}\)v\(_2 + {\alpha _3}\)v\(_3 = 0\), show that...
- 15M.2.hl.TZ0.7b: A relation \(R\) is defined on \(S\) such that \(A\) is related to \(B\) if and only if there...
- 15M.2.hl.TZ0.7a: (i) Show that \({({A^T})^{ - 1}} = {({A^{ - 1}})^T}\). (ii) Show that...
- 15M.2.hl.TZ0.5d: The matrix \(B = {A^{ - 1}}XA\) represents a reflection in the line \(y = mx\). Find the value of...
- 15M.2.hl.TZ0.5c: A triangle whose vertices have coordinates \((0,{\text{ }}0)\), \((3,{\text{ }}1)\) and...
- 15M.2.hl.TZ0.5b: Determine the matrix \(A\) which represents a rotation from the direction...
- 15M.2.hl.TZ0.5a: By considering the points \((1,{\text{ }}0)\) and \((0,{\text{ }}1)\) determine the...
- 15M.1.hl.TZ0.12b: (i) State the column rank of \(M\). (ii) Find the basis for the range of this...
- 10M.1.hl.TZ0.2a: Show that \(R\) is an equivalence relation.
- 10M.1.hl.TZ0.2b: The relationship between \(a\) , \(b\) , \(c\) and \(d\) is changed to \(ad - bc = n\) . State,...
- 12M.2.hl.TZ0.4A.a: Show that \(f\) is a bijection if \(\boldsymbol{A}\) is non-singular.
- 12M.2.hl.TZ0.4A.b: Suppose now that \(\boldsymbol{A}\) is singular. (i) Write down the relationship between...
- SPNone.1.hl.TZ0.2a: Show that the following vectors form a basis for the vector space \({\mathbb{R}^3}\)...
- SPNone.1.hl.TZ0.9a: By reducing the augmented matrix to row echelon form, (i) find the rank of the coefficient...
- SPNone.1.hl.TZ0.9b: For this value of \(k\) , determine the solution.
- SPNone.1.hl.TZ0.2b: Express the following vector as a linear combination of the above...
- SPNone.1.hl.TZ0.12a: (i) Find the matrices \({\boldsymbol{A}^2}\) and \({\boldsymbol{A}^3}\) , and verify that...
- SPNone.1.hl.TZ0.12b: (i) Suggest a similar expression for \({\boldsymbol{A}^n}\) in terms of \(\boldsymbol{A}\)...
- SPNone.1.hl.TZ0.14a: Show that any matrix of this form is its own inverse.
- SPNone.2.hl.TZ0.8a: Given that the elements of a \(2 \times 2\) symmetric matrix are real, show that (i) the...
- SPNone.2.hl.TZ0.8b: The matrix \(\boldsymbol{A}\) is given...
- 14M.1.hl.TZ0.4: The matrix M is defined by M =...
- 14M.1.hl.TZ0.10: The matrix A is given by A =...
- 14M.2.hl.TZ0.4: The matrix A is given by A =...
Sub sections and their related questions
1.1
- SPNone.1.hl.TZ0.12a: (i) Find the matrices \({\boldsymbol{A}^2}\) and \({\boldsymbol{A}^3}\) , and verify that...
- SPNone.1.hl.TZ0.12b: (i) Suggest a similar expression for \({\boldsymbol{A}^n}\) in terms of \(\boldsymbol{A}\)...
- 14M.1.hl.TZ0.10: The matrix A is given by A =...
- 15M.2.hl.TZ0.7a: (i) Show that \({({A^T})^{ - 1}} = {({A^{ - 1}})^T}\). (ii) Show that...
- 15M.2.hl.TZ0.7b: A relation \(R\) is defined on \(S\) such that \(A\) is related to \(B\) if and only if there...
- 16M.1.hl.TZ0.14a: (i) Explain why M is a square matrix. (ii) Find the set of possible values of det(M).
- 259338: This is an example question for the example test. You can delete this question.
- 18M.1.hl.TZ0.2a: Show that A4 = 12A + 5I.
- 18M.1.hl.TZ0.2b: Let B = \(\left[ {\begin{array}{*{20}{c}} 4&2 \\ 1&{ - 3} \end{array}} \right]\). Given...
- 18M.1.hl.TZ0.10c: A matrix M is said to be orthogonal if M TM = I where I is the identity. Show that Q is orthogonal.
1.2
- 10M.1.hl.TZ0.2a: Show that \(R\) is an equivalence relation.
- 10M.1.hl.TZ0.2b: The relationship between \(a\) , \(b\) , \(c\) and \(d\) is changed to \(ad - bc = n\) . State,...
- 12M.2.hl.TZ0.4A.a: Show that \(f\) is a bijection if \(\boldsymbol{A}\) is non-singular.
- 12M.2.hl.TZ0.4A.b: Suppose now that \(\boldsymbol{A}\) is singular. (i) Write down the relationship between...
- SPNone.1.hl.TZ0.14a: Show that any matrix of this form is its own inverse.
- 14M.1.hl.TZ0.10: The matrix A is given by A =...
- 15M.2.hl.TZ0.7a: (i) Show that \({({A^T})^{ - 1}} = {({A^{ - 1}})^T}\). (ii) Show that...
- 16M.1.hl.TZ0.14a: (i) Explain why M is a square matrix. (ii) Find the set of possible values of det(M).
- 17M.2.hl.TZ0.4a.i: Find an expression for det(A) in terms of \(\lambda \), simplifying your answer.
- 17M.2.hl.TZ0.4a.ii: Hence show that A is singular when \(\lambda = 1\) and find the other value of \(\lambda \) for...
1.3
- SPNone.1.hl.TZ0.9a: By reducing the augmented matrix to row echelon form, (i) find the rank of the coefficient...
- SPNone.1.hl.TZ0.9b: For this value of \(k\) , determine the solution.
- 14M.2.hl.TZ0.4: The matrix A is given by A =...
- 15M.1.hl.TZ0.12b: (i) State the column rank of \(M\). (ii) Find the basis for the range of this...
- 16M.2.hl.TZ0.8b: (i) Write down the matrices M\(_1\), M\(_2\) representing the permutations...
- 17M.2.hl.TZ0.4b.i: Explain how it can be seen immediately that B is singular without calculating its determinant.
- 17M.2.hl.TZ0.4b.ii: Determine the null space of B.
- 17M.2.hl.TZ0.4b.iii: Explain briefly how your results verify the rank-nullity theorem.
- 17M.2.hl.TZ0.4c: Prove, using mathematical induction, that B\(^n = {8^{n - 2}}\)B\(^2\) for...
1.4
- SPNone.1.hl.TZ0.9a: By reducing the augmented matrix to row echelon form, (i) find the rank of the coefficient...
- SPNone.1.hl.TZ0.9b: For this value of \(k\) , determine the solution.
- 17M.1.hl.TZ0.3a: Determine the value of \(\lambda \) and the value of \(\mu \) for which the equations are...
- 17M.1.hl.TZ0.3b: For these values of \(\lambda \) and \(\mu \), solve the equations.
- 17M.1.hl.TZ0.3c: State the rank of the matrix of coefficients, justifying your answer.
- 18M.1.hl.TZ0.8b: Explain why the equations have no unique solution when \(a\) = 3.
- 18M.1.hl.TZ0.8c: Find all the solutions to the equations when \(a\) = 3, \(b\) = 10 in the form r = s +...
1.5
- SPNone.1.hl.TZ0.2a: Show that the following vectors form a basis for the vector space \({\mathbb{R}^3}\)...
- SPNone.1.hl.TZ0.2b: Express the following vector as a linear combination of the above...
- 14M.2.hl.TZ0.4: The matrix A is given by A =...
- 15M.1.hl.TZ0.12b: (i) State the column rank of \(M\). (ii) Find the basis for the range of this...
- 16M.1.hl.TZ0.12a: Explain why the set of position vectors of points whose coordinates satisfy \(x - y - z =...
- 16M.1.hl.TZ0.12b: (i) Show that the set of position vectors of points whose coordinates satisfy...
- 17M.1.hl.TZ0.15a.i: By considering \({\alpha _1}\)v\(_1 + {\alpha _2}\)v\(_2 + {\alpha _3}\)v\(_3 = 0\), show that...
- 17M.1.hl.TZ0.15a.ii: Explain briefly why v\(_1\), v\(_2\), v\(_3\) form a basis for vectors in \({\mathbb{R}^3}\).
- 17M.1.hl.TZ0.15b.i: Show that the...
- 17M.1.hl.TZ0.15b.ii: Express the vector \[\left[ {\begin{array}{*{20}{c}} 2 \\ 8 \\ 0 \end{array}} \right]\] as a...
1.6
- 15M.1.hl.TZ0.12b: (i) State the column rank of \(M\). (ii) Find the basis for the range of this...
- 18M.1.hl.TZ0.10a.i: determine the 2 × 2 matrix P which represents a reflection in the...
- 18M.1.hl.TZ0.10a.ii: determine the 2 × 2 matrix Q which represents an anticlockwise rotation of θ about the origin.
- 18M.1.hl.TZ0.10b: Describe the transformation represented by the matrix PQ.
1.7
- 18M.1.hl.TZ0.13a: Show that the linear transformation represented by M transforms any point on the line \(y = x\)...
- 18M.1.hl.TZ0.13b: Explain what happens to points on the line \(4y + x = 0\) when they are transformed by M.
1.8
- 15M.2.hl.TZ0.5a: By considering the points \((1,{\text{ }}0)\) and \((0,{\text{ }}1)\) determine the...
- 15M.2.hl.TZ0.5b: Determine the matrix \(A\) which represents a rotation from the direction...
- 15M.2.hl.TZ0.5c: A triangle whose vertices have coordinates \((0,{\text{ }}0)\), \((3,{\text{ }}1)\) and...
- 15M.2.hl.TZ0.5d: The matrix \(B = {A^{ - 1}}XA\) represents a reflection in the line \(y = mx\). Find the value of...
- 17M.2.hl.TZ0.9a: The point \((x,{\text{ }}y)\) is rotated through an anticlockwise angle \(\alpha \) about the...
- 18M.1.hl.TZ0.4c: Write down the 2 × 2 matrices representing T3, T4 and T5.
- 18M.1.hl.TZ0.4d.i: Find the 2 × 2 matrix representing T.
- 18M.1.hl.TZ0.4d.ii: Give a geometric description of the transformation T.
1.9
- SPNone.2.hl.TZ0.8a: Given that the elements of a \(2 \times 2\) symmetric matrix are real, show that (i) the...
- SPNone.2.hl.TZ0.8b: The matrix \(\boldsymbol{A}\) is given...
- 14M.1.hl.TZ0.4: The matrix M is defined by M =...
- 16M.1.hl.TZ0.14b: (i) Find the value of \(a\). (ii) Find the eigenvalues of N. (iii) Find...
- 18M.1.hl.TZ0.13c: State the two eigenvalues of M.
- 18M.1.hl.TZ0.13d: State two eigenvectors of M which correspond to the two eigenvalues.
