MTH1004 Linear Algebra Summary Questions

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1. Determine whether the following vectors are linearly independent: (i) $u=\left(\begin{array}{c}3\\ 1\end{array}\right)$, $v=\left(\begin{array}{c}0\\ -1\end{array}\right)$, (ii) $u=\left(\begin{array}{c}2\\ -1\end{array}\right)$, $v=\left(\begin{array}{c}-1\\ -1\end{array}\right)$

2. Find a value for x so that the following vectors: $u=\left(\begin{array}{c}1\\ 1\end{array}\right)$, $v=\left(\begin{array}{c}1\\ x\end{array}\right)$ are: (i) linearly dependent. (ii) linearly independent.

3. Find two nontrivial vectors (other than ${\mathbf{e}}_1,{\mathbf{e}}_2$) that span the plane.

4. Show that the vectors: $u=\left(\begin{array}{c}-1\\ 1\end{array}\right)$, $v=\left(\begin{array}{c}1/3\\ -1/3\end{array}\right)$ are linearly dependent.

5. Determine whether the vector $u=\left(\begin{array}{c}1\\ 2\end{array}\right)$ is a linear combination of: $v=\left(\begin{array}{c}0\\ 2\end{array}\right)$, $w=\left(\begin{array}{c}1\\ 4\end{array}\right)$ If yes, find the scalars.

6. Let $\{e_1,e_2,e_3\}$ be the standard vectors in ${\mathbb{R}}^3$. Simplify the sets: (i) $\text{span}(e_1,e_1+e_2,e_1+e_2+e_3)$ (ii) $\text{span}(e_1,-2e_2,e_1+e_2-e_3)$ (iii) $\text{span}(e_1+e_2,e_2,e_2-3e_1)$

7. If $u,v,w$ are linearly independent vectors, determine whether $u$ can be expressed as a linear combination of $v,w$.

8. Find the span of the following vectors: (i) $u=\left[\begin{array}{c}2\\ 0\\ 3\end{array}\right]$, $v=\left[\begin{array}{c}0\\ 1\\ 1\end{array}\right]$ (ii) $u=\left[\begin{array}{cccc}0& -2& 0& 5\end{array}\right]$, $v=\left[\begin{array}{cccc}3& 1& 1& 0\end{array}\right]$ (iii) $u=\left[\begin{array}{cccc}1& -1& 0& 2\end{array}\right]$, $v=\left[\begin{array}{cccc}0& -1/2& 3/2& 1\end{array}\right]$, $w=\left[\begin{array}{cccc}-1& 0& 1& 0\end{array}\right]$

9. Find a value for $c$ so that the following vectors: (i) $u=\left(\begin{array}{c}1\\ 1\\ -c\end{array}\right)$, $v=\left(\begin{array}{c}1\\ 0\end{array}\right)$ (ii) $u=\left[\begin{array}{ccc}1& c& 3\end{array}\right]$, $v=\left[\begin{array}{ccc}1/3& -1/3& 1\end{array}\right]$, are linearly independent.

10. Check whether the following vectors in ${\mathbb{R}}^3$ are linearly independent: $u=\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right]$, $v=\left[\begin{array}{c}-1\\ 1\\ 0\end{array}\right]$, $w=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]$, and find $\text{span}(u,v,w)$.

11. Find the dot product, the length, and the angle between the vectors: (i) $u=\left(\begin{array}{c}1\\ -1\end{array}\right)$, $v=\left(\begin{array}{c}0\\ 3\end{array}\right)$, (ii) $u=\left[\begin{array}{ccc}0& 1/2& -1/2\end{array}\right]$, $v=\left[\begin{array}{ccc}1/3& 2/3& 2/3\end{array}\right]$, (iii) $u=\left[\begin{array}{cccc}0& -1/2& 2& 2\end{array}\right]$, $v=\left[\begin{array}{cccc}1& -1& 1& -1\end{array}\right]$. Verify the Cauchy-Schwarz inequality and the Triangle inequality.

12. Find the solution set which corresponds to the following row echelon forms: (i) $\left[\begin{array}{cccccc}1& -2& -1& 0& 2& 1\\ 0& 0& 0& 0& 0& 0\end{array}\right]$ (ii) $\left[\begin{array}{cccccc}1& 0& 1& 2& 0& 3\\ 0& 0& 0& 0& 1& -1\end{array}\right]$

13. Solve the system with 3 equations and 4 unknowns using Gauss-Jordan elimination: $\{\begin{array}{l}{x}_1+2x_2-2x_3+2x_4=1\\ 2x_1-4x_2+4x_3+6x_4=0\\ x_1+2x_2-2x_3+3x_4=0\end{array}$ and write its set of solutions.

14. Let $v_1=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],v_2=\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right],v_3=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$ vectors in ${\mathbb{R}}^3$. Using Gauss-Jordan elimination: (i) Show that the vectors are linearly dependent. (ii) Determine whether the vector $u=\left[\begin{array}{c}0\\ 1\\ 1\end{array}\right]$ belongs to the span of the vectors $v_1,v_2,v_3$.

15. Consider the following matrices $B_1=\left[\begin{array}{ccc}-1& 0& 1\\ 0& 2& 1\\ 0& 0& 5\\ 0& 0& -1\end{array}\right],B_2=\left[\begin{array}{cccc}1& 2& 0& 5\\ 0& 0& 0& 1\\ 0& 0& 0& 0\end{array}\right],B_3=\left[\begin{array}{cccc}1& -1& 0& 1\\ 0& 0& 0& 0\end{array}\right],B_4=\left[\begin{array}{ccccc}1& 0& 0& -2& 5\\ 0& 1& 0& 9& -2\\ 0& 0& 1& 4& 3\end{array}\right]$ Find the matrices which are in row-echelon form. Which of those matrices are in reduced row-echelon form?

16. Find $A+B$ and $A\cdot B$, when $A=\left[\begin{array}{ccc}1& -1& 1\\ 0& 3& 1\\ 1& -2& 5\end{array}\right]$, $B=\left[\begin{array}{ccc}1& 0& 0\\ -1& 2& 0\\ 3& -2& 3\end{array}\right]$. Do the matrices A, B commute ($AB=BA$)?

17. Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$. Find conditions for $a,b,c,d$ which make A: (a) A diagonal matrix (b) A symmetric matrix (c) An upper triangular matrix (d) A skew-symmetric matrix ($A^T=-A$).

18. Let $Ax=0$ be a homogeneous system with 3 equations and 3 unknowns. Find the rank of the matrix A, when the system has: (a) a unique solution. (b) infinite solutions of the form $x=2z,y=-z,z$ in $\mathbb{R}$.

19. Determine the rank of following matrices $A=\left[\begin{array}{cc}2& -1\\ -1& 1/2\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$ and of their matrix powers $A^2$ and $B^2$. (a) Is $(A-B)(A+B)=A^2-B^2$ true for any $n\times n$ A, B matrices? Justify your answer. (b) Show that $A{A}^T$ is a symmetric matrix.

20. Let $u,v$ be two linearly independent vectors in ${\mathbb{R}}^2$ and $A$ the matrix with columns the vectors $u,v$. What is the rank of A?

21. Find the inverse of the following $2\times 2$ matrices: $A=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right),B=\left(\begin{array}{cc}a& -b\\ b& a\end{array}\right)$ using the standard formula for the inverse of $2\times 2$ matrices: $A^{-1}=\frac{1}{ad-bc}\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)$.

22. Find the inverse of the matrix $B=\left(\begin{array}{ccc}3& 1& 2\\ -1& 1& 2\\ 1& 3& 0\end{array}\right)$ using Gauss-Jordan elimination and its rank.

23. Prove that: (a) If $A$ is invertible and $AB=O$, then $B=O$. (b) If $A$ is invertible and $AB=AC$, then $B=C$. (c) If $A$ satisfies the matrix equation $A^2-2A+I=O$, then $A$ is invertible and find its inverse. (d) If $A$ satisfies the matrix equation $A^3+\alpha A^2+\beta A-I=O$, then $A$ is invertible and find its inverse. (a) Compute the determinant of the following matrices: $A=\left(\begin{array}{ccc}2& -3& 2\\ 4& 1& 0\\ 0& 0& 6\end{array}\right),B=\left(\begin{array}{ccc}-2& 1& 0\\ 1& -2& 1\\ 0& 1& -2\end{array}\right)$. (b) Find the determinant of the matrix $A^n$.

24. Using the determinant properties (and not cofactor expansions), show that: $\det (P^{-1}AP)=\det A$ $\det (P^{T}AP)=\det A(\det P{)}^2$ (c) If $\det A=1$ and $ABAT=B^2$, then $\det B=0$ or 1.

25. Use Cramer’s rule to solve the following linear systems: $\{\begin{array}{l}2x+y-z=1\\ -x+2y+z=1\\ x-y+2z=1\end{array}$, $\{\begin{array}{l}x+ay+az=0\\ x-y+az=0\\ x+y+z=0\end{array}$ For which $a$ values the second system has unique solution?

26. Show that $S=\{\left(\begin{array}{c}x\\ x-y\\ x+2y\end{array}\right):x,y\in \mathbb{R}\}$ is the vector subspace of ${\mathbb{R}}^3$ and find a basis for $S$.

27. Which of the following sets are vector subspaces of ${\mathbb{R}}^3$ and why. (a) $S_1=\{x\in {\mathbb{R}}^3:x_1=x_2=x_3\}$ (b) $S_2=\{x\in {\mathbb{R}}^3:x_1+x_2+x_3=0\}$ (c) $S_3=\{x\in {\mathbb{R}}^3:x_1\ge 0\}$

28. Find a basis for the set of solutions of the following linear system: $n{x}_1+x_2+3x_3-x_4=0$, $x_2+4x_3+x_4=0$

29. Check whether the following vectors form a basis for ${\mathbb{R}}^3$: $u=\left(\begin{array}{c}1\\ 1\\ 2\end{array}\right)$, $v=\left(\begin{array}{c}2\\ 1\\ -1\end{array}\right)$, $w=\left(\begin{array}{c}1\\ 1\\ -2\end{array}\right)$

30. Find $\text{col}(A)$, $\text{row}(A)$, and $\text{null}(A)$ for the matrix: $A=\left(\begin{array}{ccc}1& 1& -1\\ -1& 1& 1\\ 1& -1& 1\end{array}\right)$. Find a basis for each for the above vector subspaces and determine their dimension. Verify the Rank Theorem: $\text{rank}(A)+\text{nullity}(A)=\text{dim}({\mathbb{R}}^3)$.

31. Find the eigenvalues and eigenvectors of the following matrix: $B=\left(\begin{array}{ccc}1& -12& 4\end{array}\right)$. Show that its determinant equals to ${\lambda }_1{\lambda }_2$.

32. Prove that any $2\times 2$ matrix of the form: $A=\left(\begin{array}{ccc}a& bb& a\end{array}\right)$ where $a\ne b$, $a$, $b$ in $\mathbb{R}$, has two real eigenvalues, which are: $a+b$, $a-b$.

33. Diagonalise the matrix: $A=\left(\begin{array}{ccc}1& 0& 1\\ 0& 1& 1\\ 1& 1& 0\end{array}\right)$.

34. Find $B^n$ when $B=\left(\begin{array}{cc}0& 2\\ 1& 1\end{array}\right)$.

35. Show that $A=\left(\begin{array}{cc}1& -1\\ 1& 1\end{array}\right)$ is non-diagonalisable. The matrix $B=\left(\begin{array}{cc}2& -1\\ 0& 2\end{array}\right)$ has two equal eigenvalues ${\lambda }_{1,2}=\lambda =2$. What is the algebraic and what is the geometric multiplicity of $\lambda$? Is $B$ diagonalisable?

36. Consider the following transformations: (i) $T:{\mathbb{R}}^2\to {\mathbb{R}}^3$ $T\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}-y\\ x-y\\ 2x-y\end{array}\right]$ (ii) $T:{\mathbb{R}}^3\to {\mathbb{R}}^3$ $T\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}x+2y\\ y-2z\\ x+y+z\end{array}\right]$ (iii) $T:{\mathbb{R}}^3\to {\mathbb{R}}^2$ $T\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}x-y+z\\ x+y-z\end{array}\right]$

37. Show that these transformations are linear.

38. Find a basis for the kernel and the range of the linear transformations (in Problem 1). What is their nullity and rank?

39. Which of the linear transformations (in Problem 1) are one-to-one and which are onto? Is any of them an isomorphism? Justify your answer.

40. Let $\{e_1,e_2,e_3\}$ be the standard vectors in ${\mathbb{R}}^3$. Determine whether $\text{span}(e_1,-2e_2,e_1+e_2-e_3)={\mathbb{R}}^3$.

41. Find the span of the following vectors: (i) $u=\left[\begin{array}{c}0\\ -2\\ 0\\ 5\end{array}\right]$, $v=\left[\begin{array}{c}3\\ 1\\ 1\\ 0\end{array}\right]$, (ii) $u=\left[\begin{array}{c}1\\ -1\\ 0\\ 2\end{array}\right]$, $v=\left[\begin{array}{c}0\\ -\frac{1}{2}\\ \frac{3}{2}\\ 1\end{array}\right]$, $w=\left[\begin{array}{c}-1\\ 0\\ 1\\ 0\end{array}\right]$

42. Find a value for $c$ so that the following vectors: (i) $u=\left(\begin{array}{c}1\\ 1\end{array}\right)$, $v=\left(\begin{array}{c}1\\ 0\end{array}\right)$ are linearly independent.

43. Find the dot product, the length and the angle between the vectors: $u=\left[\begin{array}{c}0\\ \frac{1}{2}\\ -\frac{1}{2}\end{array}\right]$, $v=\left[\begin{array}{c}\frac{1}{3}\\ \frac{2}{3}\\ \frac{2}{3}\end{array}\right]$. Verify the Cauchy-Schwarz inequality and the triangle inequality.

44. Using Gauss-Jordan elimination: (i) Determine whether the vector $u=\left[\begin{array}{c}9\\ 0\\ 4\end{array}\right]$ belongs to the span of the following vectors: $v=\left[\begin{array}{c}1\\ 2\\ 4\end{array}\right]$, $w=\left[\begin{array}{c}2\\ -1\\ -1\end{array}\right]$, $z=\left[\begin{array}{c}-3\\ 1\\ 1\end{array}\right]$. (ii) Show that the vectors $u=\left[\begin{array}{c}1\\ -1\\ 2\end{array}\right]$, $v=\left[\begin{array}{c}-3\\ 2\\ 4\end{array}\right]$, $w=\left[\begin{array}{c}-2\\ 1\\ 6\end{array}\right]$ are linearly dependent and simplify span(u, v, w) in terms of u, v, w.

45. For what $k$ values will the systems: (a) $nkx+2y=3$, $2x-4y=-6$, (b) $nx+ky=1$, $kx+y=1$ have: (i) infinitely many solutions, (ii) no solutions, and (iii) a unique solution? For (i) write the set of solutions.

46. Problem 1: Consider the following matrices: $A_1=\left(\begin{array}{ccc}2& 3& -1\\ 0& 2& -1\end{array}\right)$, $A_2=\left(\begin{array}{ccc}1& 0& -1\\ 3& 0& -1\\ -1& 0& 4\end{array}\right)$, $A_3=\left(\begin{array}{ccc}1& 0& -2\\ 0& 1& 2\\ 0& 0& 0\end{array}\right)$, $A_4=\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)$, $B_1=\left(\begin{array}{cccc}1& -1& -1& 0\\ 1& -1& 0& 0\\ 1& 0& -1& 0\\ 0& -1& 0& -1\end{array}\right)$, $B_2=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 1\end{array}\right)$, $B_3=\left(\begin{array}{ccc}1& 4& 0\\ 1& 0& 0\end{array}\right)$, $B_4=\left(\begin{array}{cccc}2& 0& 1& 3\\ -8& 0& 0& -1\\ -1& 3& 0& 0\\ 4& 7& 7& 0\end{array}\right)$. Find the matrices which are in row-echelon form and compute their rank. Which of those matrices are in reduced row-echelon form?

47. Problem 2: Show that: (i) $A{A}^T$ and (ii) $A+A^T$ are symmetric matrices.

48. Problem 3: Find $x$ when: $\left(\begin{array}{cccc}2& -1& 1& 3\end{array}\right)\left(\begin{array}{c}x\\ 3\\ 2\\ -1\end{array}\right)=\left(\begin{array}{c}-6\\ 7\\ 4\\ 0\end{array}\right)$.

49. Problem 1: (a) If $A$ is invertible, prove that a homogeneous system $Ax=0$ has unique solution the zero solution $x=0$.

50. Problem 1: (b) If $A$ is invertible, what is the rank of $A$? Justify your answer.

51. Problem 1: (c) Simplify the expression $(A^{-1}{B}^2{C}^{-2}{)}^{-1}$.

52. Problem 1: (d) Prove that, if $A$ satisfies the matrix equation $4A^3-A^2-2I=O$, then $A$ is invertible.

53. Problem 1: (e) Solve the matrix equation $A^{-1}X+B+3A=O$ in terms of the matrix $X$.

54. Find the inverse of the matrix: (a) $A=\left(\begin{array}{cc}1& -2\\ 5& 7\end{array}\right)$ using the standard formula.

55. Find the inverse of the matrix: (b) $R=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$ using the standard formula.

56. Find the inverse of the matrix: (c) $C=\left[\begin{array}{ccc}1& 1& 2\\ 1& 2& 1\\ 2& 1& 1\end{array}\right]$ using Gauss-Jordan elimination. By using its inverse $C^{-1}$, solve the system $Cx=u$, where $u=\left[\begin{array}{c}2\\ 1\\ 0\end{array}\right]$.

57. Prove that every line through the origin in ${\mathbb{R}}^3$ is a subspace of ${\mathbb{R}}^3$.

58. Do the vectors $\left(\begin{array}{c}0\\ 1\\ 1\end{array}\right),\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right),\left(\begin{array}{c}1\\ 1\\ 0\end{array}\right)$ form a basis for ${\mathbb{R}}^3$? And $\left(\begin{array}{c}0\\ 1\\ 1\end{array}\right),\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right),\left(\begin{array}{c}-2\\ 3\\ 1\end{array}\right)$?

59. Suppose that $S$ consists of all points in ${\mathbb{R}}^2$ that are on the x-axis or the y-axis (or both). ($S$ is called the union of the two axes.) Is $S$ a subspace of ${\mathbb{R}}^2$? Why or why not?

60. If $A$ and $B$ are $n\times n$ matrices of rank $n$, prove that the matrix $AB$ has rank $n$ (hint: use the fact that $A$ and $B$ are invertible).

61. Show that the set $S=\{x\in {\mathbb{R}}^3:x_1\le 0\text{ and }{x}_2\ge 0\}$ is not a vector space by showing that at least one of the vector space axioms is not satisfied.

62. Show that the set of all upper triangular $2\times 2$ matrices is a vector space.