Linear Algebra Summary Questions

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1. Determine whether the following vectors are linearly independent:

   1. $\vec{u}=\left[\begin{array}{c}3\\ 1\end{array}\right],\vec{v}=\left[\begin{array}{c}0\\ -1\end{array}\right]$.
   2. $\vec{u}=\left[\begin{array}{c}2\\ -1\end{array}\right],\vec{v}=\left[\begin{array}{c}-1\\ -1\end{array}\right]$.

2. Find a value for $x$ so that the following vectors are linearly dependent, and another value for which they are linearly independent: $\vec{u}=\left[\begin{array}{c}1\\ 1\end{array}\right],\vec{v}=\left[\begin{array}{c}1\\ x\end{array}\right]$.

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

4. Show that the following vectors are linearly dependent: $\vec{u}=\left[\begin{array}{c}-1\\ 1\end{array}\right],\vec{v}=\left[\begin{array}{c}\frac{1}{3}\\ -\frac{1}{3}\end{array}\right]$.

5. Determine whether the vector $\vec{u}=\left[\begin{array}{c}1\\ 2\end{array}\right]$ is a linear combination of $\vec{v}=\left[\begin{array}{c}0\\ 2\end{array}\right]$ and $\vec{w}=\left[\begin{array}{c}1\\ 4\end{array}\right]$. If so, find the scalars.

6. Let $\{{\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3\}$ be the standard vectors in ${\mathbb{R}}^3$. Determine whether $\text{span}({\mathbf{e}}_1,-2{\mathbf{e}}_2,{\mathbf{e}}_1+{\mathbf{e}}_2-{\mathbf{e}}_3)={\mathbb{R}}^3$.

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

8. Find a value for $c$ so that the following vectors are linearly independent: $\mathbf{u}=\left[\begin{array}{c}1\\ 1-c\end{array}\right],\mathbf{v}=\left[\begin{array}{c}1\\ 0\end{array}\right]$.

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

10. Let $\{{\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3\}$ be the standard vectors in ${\mathbb{R}}^3$. Simplify the sets:

    1. $\text{span}({\mathbf{e}}_1,{\mathbf{e}}_1+{\mathbf{e}}_2,{\mathbf{e}}_1+{\mathbf{e}}_2+{\mathbf{e}}_3)$.
    2. $\text{span}({\mathbf{e}}_1,-2{\mathbf{e}}_2,{\mathbf{e}}_1+{\mathbf{e}}_2-{\mathbf{e}}_3)$.
    3. $\text{span}({\mathbf{e}}_1+{\mathbf{e}}_2,{\mathbf{e}}_2,{\mathbf{e}}_2-3{\mathbf{e}}_1)$.

11. If $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$ are linearly independent vectors, determine whether $\mathbf{u}$ can be expressed as a linear combination of $\mathbf{v}$ and $\mathbf{w}$.

12. Find the span of the following vectors:

    1. $\mathbf{u}=\left[\begin{array}{c}2\\ 0\\ 3\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{c}0\\ 1\\ 1\end{array}\right]$.
    2. $\mathbf{u}=\left[\begin{array}{c}0\\ -2\\ 0\\ 5\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{c}3\\ 1\\ 1\\ 0\end{array}\right]$.
    3. $\mathbf{u}=\left[\begin{array}{c}1\\ -1\\ 0\\ 2\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{c}0\\ -\frac{1}{2}\\ \frac{3}{2}\\ 1\end{array}\right]$ and $\mathbf{w}=\left[\begin{array}{c}-1\\ 0\\ 1\\ 0\end{array}\right]$.

13. Find a value for $c$ so that the following vectors are linearly independent:

    1. $\mathbf{u}=\left[\begin{array}{c}1\\ 1-c\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{c}1\\ 0\end{array}\right]$.
    2. $\mathbf{u}=\left[\begin{array}{c}1\\ c\\ 3\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{c}\frac{1}{3}\\ -\frac{1}{3}\\ 1\end{array}\right]$.

14. Check whether the following vectors in ${\mathbb{R}}^3$ are linearly independent, and then find the span of the three vectors:

$$\mathbf{u}=\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right],\mathbf{v}=\left[\begin{array}{c}-1\\ 1\\ 0\end{array}\right],\mathbf{w}=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]$$

15. Find the dot product, the length, and the angle between the vectors:

    1. $\mathbf{u}=\left[\begin{array}{c}1\\ -1\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{c}0\\ 3\end{array}\right]$.
    2. $\mathbf{u}=\left[\begin{array}{c}0\\ \frac{1}{2}\\ -\frac{1}{2}\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{c}\frac{1}{3}\\ \frac{2}{3}\\ \frac{2}{3}\end{array}\right]$.
    3. $\mathbf{u}=\left[\begin{array}{c}0\\ -\frac{1}{2}\\ 2\\ 2\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{c}1\\ -1\\ 1\\ -1\end{array}\right]$.
    4. Then, verify the Cauchy-Schwarz inequality and the Triangle inequality.

16. Complete the following questions using Gauss-Jordan elimination:

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

17. Find a value of $k$ to satisfy the following constraints for the systems below:

    1. Infinitely many solutions: write the set of solutions.
    2. No solutions.
    3. A unique solution.

kx+2y=3 \\ 2x-4y=-6 \end{cases}\quad,(b)\quad\begin{cases} x+ky=1 \\ kx+y=1 \end{cases}$$ 18. Find the solution set which corresponds to the following row echelon forms: 1. $\begin{bmatrix}1 & -2 & -1|0 \\ 0 & 2 & 1|0\end{bmatrix}$, 2. $\begin{bmatrix}1 & 0 & 1 & 2|0 \\ 0 & 3 & -1 & -1|0\end{bmatrix}$. 19. Solve the system with three equations and four unknowns using Gauss-Jordan elimination, and write its set of solutions: $$ \begin{cases} x_{1}+2x_{2}-2x_{3}+2x_{4}=1 \\ 2x_{1}-4x_{2}+4x_{3}+6x_{4}=0 \\ x_{1}+2x_{2}-2x_{3}+2x_{4}=0 \end{cases}

20. Let ${\mathbf{v}}_1=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],{\mathbf{v}}_2=\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right],{\mathbf{v}}_3=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$ be vectors in ${\mathbb{R}}^3$. Using Gauss-Jordan elimination:

    1. Show that the vectors are linearly dependent.
    2. Determine whether the vector $\mathbf{u}=\left[\begin{array}{c}0\\ 1\\ 1\end{array}\right]$ belongs to the span of the vectors ${\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3$.

21. Considering the following matrices, denote which are in row-echelon form and which are in reduced row-echelon form:

    1. $B_1=\left[\begin{array}{ccc}-1& 0& 1\\ 0& 2& 1\\ 0& 0& 5\\ 0& 0& -1\end{array}\right]$,
    2. $B_2=\left[\begin{array}{cccc}1& 2& 0& 5\\ 0& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right]$,
    3. $B_3=\left[\begin{array}{cc}1& -1\\ 0& 1\\ 0& 0\end{array}\right]$,
    4. $B_4=\left[\begin{array}{ccccc}1& 0& 0& -2& 5\\ 0& 1& 0& 9& -2\\ 0& 0& 1& 4& 3\end{array}\right]$.

22. Consider the following matrices, find which are in row-echelon form and compute their rank, then, find which of those matrices are in reduced row-echelon form:

$$A_1=\left[\begin{array}{ccc}2& 3& -1\\ 0& 2& -1\end{array}\right],A_2=\left[\begin{array}{cccc}1& 0& -1& 3\\ 0& -1& -1& 0\\ 0& 0& 0& 4\end{array}\right],A_3=\left[\begin{array}{ccc}1& 0& -2\\ 0& 1& 2\\ 0& 0& 0\end{array}\right],A_4=\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right],$$ $$B_1=\left[\begin{array}{ccc}1& -1& -1\\ 0& 1& -1\\ 0& 0& 1\\ 0& 0& -1\end{array}\right],B_2=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],B_3=\left[\begin{array}{cc}1& 4\\ 0& 1\\ 0& 0\end{array}\right],B_4=\left[\begin{array}{ccccc}2& 0& 1& 3& -8\\ 0& 0& -1& -1& 3\\ 0& 0& 4& 7& 7\end{array}\right].$$

23. Show that both $A{A}^T$ and $A+A^T$ are symmetric matrices.

24. Find $x$ when:

$$\left[\begin{array}{cc}2& -1\\ 1& 3\end{array}\right]\left[\begin{array}{cc}x& 3\\ 2& -1\end{array}\right]=\left[\begin{array}{cc}-6& 7\\ 4& 0\end{array}\right]$$

25. Find $A+B$ and $A\cdot B$, then answer whether or not the matrices commute, where

$$A=\left[\begin{array}{ccc}1& -1& 1\\ 0& 3& 1\\ 1& -2& 5\end{array}\right],B=\left[\begin{array}{ccc}1& 0& 0\\ -1& 2& 0\\ 3& -2& 3\end{array}\right]$$

26. Find conditions for $a,b,c,d$ which make $A$: a diagonal matrix, a symmetric matrix, an upper triangle matrix, and a skew-symmetric matrix (such that $A^T=-A$), where

$$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$$

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

28. Determine the rank of the following matrices, and of their matrix powers $A^2$ and $B^2$:

$$A=\left[\begin{array}{cc}2& -1\\ -1& \frac{1}{2}\end{array}\right]\text{and}B=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$$

29. Answer the following, where $A,B$ are any $n\times n$ matrices:

    1. Is $(A-B)(A+B)=A^2-B^2$ true?
    2. Show that $A{A}^T$ is a symmetric matrix.

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

31. If $A$ is invertible, prove that a homogenous system $A\mathbf{x}=0$ has a unique solution: the zero solution $\mathbf{x}=0$.

32. If $A$ is invertible, what is the rank of $A$? Justify your answer.

33. Simplify the expression $(A^{-1}{B}^2{C}^{-2}{)}^{-1}$.

34. Prove that if $A$ satisfies the matrix equation $4A^3-A^2-2I=O$, then $A$ is invertible.

35. Solve the matrix equation $A^{-1}X+B+3A=O$ in terms of the matrix $X$.

36. Find the inverses of the following matrices, where any $2\times 2$ matrices are completed with the standard formula and any $3\times 3$ are with Gauss-Jordan elimination:

    1. $A=\left[\begin{array}{cc}1& -2\\ 5& 7\end{array}\right]$
    2. $R=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$
    3. $C=\left[\begin{array}{ccc}1& 1& 2\\ 1& 2& 1\\ 2& 1& 1\end{array}\right]$
       1. By then using the inverse $C^{-1}$, solve the system $C\mathbf{x}=\mathbf{u}$, where $\mathbf{u}=\left[\begin{array}{c}2\\ 1\\ 0\end{array}\right]$.

37. Find the inverse of the following $2\times 2$ matrices, using the standard formula:

$$A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]$$

38. Find the inverse of the following matrix using Gauss-Jordan elimination and its rank:

$$B=\left[\begin{array}{ccc}3& 1& 2\\ -1& 1& 2\\ 2& 1& 3\end{array}\right]$$

39. Prove that…

    1. If $A$ is invertible and $AB=O$, the $B=O$.
    2. If $A$ is invertible and $AB=AC$, then $B=C$.
    3. If $A$ satisfies the matrix equation $A^2-2A+I=O$, then $A$ is invertible and find its inverse.
    4. If $A$ satisfies the matrix equation $A^3+\alpha A^2+\beta A-I=O$, then $A$ is invertible and find its inverse.

40. Compute the determinant of the following matrices, then find the determinant of the matrix $A^n$:

$$A=\left[\begin{array}{ccc}2& -3& 2\\ 4& 1& 0\\ 0& 0& 6\end{array}\right],B=\left[\begin{array}{ccc}-2& 1& 0\\ 1& -2& 1\\ 0& 1& -2\end{array}\right]$$

41. Using the determinant properties (and not cofactor expansions), show that:

    1. $\det (P^{-1}AP)=\det A$,
    2. $\det (P^{T}AP)=\det A(\det P{)}^2$.,
    3. If $\det A=1$ and $AB{A}^T=B^2$, then $\det B=\{\begin{array}{l}0\\ 1\end{array}$.

42. Use Cramer’s rule to solve the following linear systems and then determine which $a$ values the second system has a unique solution:

$$\{\begin{array}{l}2x+y-z=1\\ -x+2y+z=1\\ x-y+2z=1\end{array},\{\begin{array}{l}x+ay+az=0\\ x-y+az=0\\ x+y+z=0\end{array}$$