Linear Algebra Expert Q&A

시스템에 해가 없거나, 정확히 하나의 해 또는 무한히 많은 해가 없는 의 값을 결정한다. $x + 2y - 3z = 4,$ $3x - y + 5z = 2,$ $4x + y + (a^2 - 14)z = a + 2.$

(a) 증명: AB와 BA가 모두 정의되어 있다면 AB와 BA는 정방행렬이다.

(b) 증명: A가 m x n 행렬이고 A(BA)가 정의되어 있다면 B는 n x m 행렬이다.

Determine which of the following pairs of functions are linearly independent:

1. $f(\theta) = \cos(3\theta),\;g(\theta) =16 \cos^3 \theta-12\cos\theta$

2. $f(t) = e^{\lambda t} \cos\mu t,\; g(t) = e^{\lambda t} \sin \mu t$

3. $f(x,y) = 2x-4y-12,\; g(x,y) =-3x + 6y + 18$

4. $f(t)=5t^2 + 35t,\; g(t)= 5t^2-35$

5. $f(t) = 17t^3,\; g(t) = e^t$

6. $f(t) = 3t, g(t)=|t|$

7. $f(x) = x^2,\; g(x) = 4x^2$

8. $f(x)=x^3,\; g(x)=|x|^3$

9. $f(x)=e^{5x},\; g(x)=e^{5(x-3)}$

Although the growth equation $y^{\prime}=k y$ is simple, it is not easy to approximate numerically, particularly over intervals [0, a] for large a. Compare the accuracy of different numerical methods by solving the $IVP y^{\prime}=y, y(0)=1,$ and evaluating the solution at t = 1. The exact value of y(1) is e, so all methods can be compared against this value. Try step sizes of 0.1, 0.5, and 1. Although a step size of 1 is enormous, you may be surprised at the accuracy of certain other methods even so- comment on why this might be so.

Read your documentation on how to enter matrices and how to produce reduced row echelon forms. Check your understanding of these commands by finding the reduced row echelon form of the matrix

$$\left[\begin{array}{rrrrr}{2} & {-3} & {1} & {0} & {4} \\ {1} & {1} & {2} & {2} & {0} \\ {3} & {0} & {-1} & {4} & {5} \\ {1} & {6} & {5} & {6} & {-4}\end{array}\right]$$

.

Determine if the subset of $\mathbb{R}^3$ is a subspace of $\mathbb{R}^3$:

$$\{[9x,\;-5x,\;-2x]^T : x\;\text{ arbitrary\; number}\}.$$

Determine if the subset of $\mathbb{R}^3$ is a subspace of $\mathbb{R}^3$:

$$\{[x,\;x+3,\;x+6]^T : x \;\text{arbitrary\; number}\}.$$

Use your technology utility to compute the components of $\mathbf{u}=(7.1,-3)-5(\sqrt{2}, 6)+3(0, \pi)$ to five decimal places.

(a) Find the sine and cosine of the angle between the vectors $\mathbf{u}=(1,-2,4,1)$ and $\mathbf{v}=(7,4,-3,2)$. (b) Find the angle between the vectors in part (a).

Use a software program or a graphing utility to solve the system of linear equations.

$$\begin{align*} \frac{1}{2}x_1-\frac{3}{7}x_2+\frac{2}{9}x_3 &= \frac{349}{630} \\ \frac{2}{3}x_1+\frac{4}{9}x_2-\frac{2}{5}x_3 &= -\frac{19}{45} \\ \frac{4}{5}x_1-\frac{1}{8}x_2+\frac{4}{3}x_3 &= \frac{139}{150} \\ \end{align*}$$

Find the eigenvalues $\lambda_1 < \lambda_2 < \lambda_3$ and corresponding eigenvectors of the matrix

$$A=\left[\begin{array}{ccc} -5 & 15 & -120\\ 0 & 0 & 20\\ 0 & 0 & 4 \end{array}\right].$$

Let u be the vector in $R^{100}$ whose $i \mathrm{th}$ component is i, and let v be the vector in $R^{100}$ whose $i \mathrm{th}$ component is $1 /(i+1)$. Evaluate the dot product $\mathbf{u} \cdot \mathbf{v}$ by first writing it in sigma notation.

Rotate axes to identify the graph of the equation and write the equation in standard form.

$$x y=1$$

list $\mu5$ and $\mu6$.

$$A=\left[\begin{array}{ll}5 & 2 \\ 2 & 2\end{array}\right], x_0=\left[\begin{array}{l}1 \\ 0\end{array}\right]$$

Determine if the subset of $\mathbb{R}^3$ is a subspace of $\mathbb{R}^3$:

$$\{[9x-7y,\;-5x-4y,\;-2x+4y]^T : x,y\;\text{arbitrary\; numbers}\}.$$