# Linear algebra questions and answers

Recent questions in Linear algebra
Forms of linear equations

### Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row echelon form. Solve the system. Assume that the variables are named $$x_1,x_2,…x ,…$$ from left to right. $$\begin{bmatrix}1&0&0&-3 \\0&1&0&0\\0&0&1&7 \end{bmatrix}$$

Vectors and spaces

### Identify the surface with the given vector equation. r(s,t)=$$\displaystyle{\left({s},{t},{t}^{{2}}-{s}^{{2}}\right)}$$ eliptic cylinder circular paraboloid hyperbolic paraboloid plane circular cylinder

Forms of linear equations

### A bank loaned out $11,000, part of it at the rate of 7% annual interest, and the rest at 9% annual interest. The total interest earned for both loans was$860.00. How much was loaned at each rate? ___ was loaned at 7% and ___ was loaned at 9%.

Vectors and spaces

### Find a unit vector that has the same direction as the given vector. 8i-j+4k

Vectors and spaces

### An idealized velocity field is given by the formula $$\displaystyle{V}={4}{t}\xi-{2}{t}^{{2}}{y}{j}+{4}{x}{z}{k}$$ Is this flow field steady or unsteady? Is it two or three dimensional? At the point $$\displaystyle{\left({x},{y},{z}\right)}={\left(-{1},{1},{0}\right)}$$, compute (a) the acceleration vector.

Vectors and spaces

### Find a unit vector that is orthogonal to both u = (1, 0, 1) and v = (0, 1, 1).

Matrix transformations

### Find an orthogonal basis for the column space of each matrix. $$\begin{bmatrix}3&-5&1\\1&1&1\\-1&5&-2\\3&-7&-8\end{bmatrix}$$

Matrix transformations

### Consider the linear system $$\vec{y}'=\begin{bmatrix}-3 & -2 \\ 6 & 4 \end{bmatrix}\vec{y}$$ a) Find the eigenvalues and eigenvectors for the coefficient matrix $$\lambda_{1}1,\ \vec{v}_{1}=\begin{bmatrix} -1 \\ 2 \end{bmatrix}$$ and $$\lambda_{2}=0,\ \vec{v}_{2}=\begin{bmatrix} -2 \\ 3 \end{bmatrix}$$ b) For each eigenpair in the previos part, form a solution of $$\displaystyle\vec{{{y}}}'={A}\vec{{{y}}}$$. Use t as the independent variable in your answers. $$\vec{y}_{1}(t)=\begin{bmatrix} & \\ & \end{bmatrix}$$ and $$\vec{y}_{2}(t)=\begin{bmatrix} -2 \\ 3 \end{bmatrix}$$ c) Does the set of solutions you found form a fundamental set (i.e., linearly independent set) of solution? No, it is not a fundamental set.

Forms of linear equations

### Form (a) the coefficient matrix and (b) the augmented matrix for the system of linear equations. \left{\begin{array}{c} 7 x-5 y=11 x-y=-5 \end{array}\right.

Matrix transformations

### Use the definition of Ax to write the matrix equation as a vector equation, or vice versa. $$\left[\begin{array}{c} 7 & -3 \\ 2 & 1 \\ 9 & -6 \\ -3 & 2\end{array}\right] \left[\begin{array}{c} -2 \\ -5\end{array}\right]=\left[ \begin{array}{c}1 \\ -9 \\ 12 \\ -4\end{array}\right]$$

Matrix transformations

### Consider the $$\displaystyle{3}\times{3}$$ matrices with real entrices. Show that the matrix forms a vector space over R with respect to matrix addition and matrix multiplication by scalars?

Forms of linear equations

### Determine whether each first-order differntial equation is separable, linear, both or nether: a) $$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}+{e}^{{{x}}}{y}={x}^{{{2}}}{y}^{{{2}}}$$ b) $$\displaystyle{y}+{\sin{{x}}}={x}^{{{3}}}{y}'$$ c) $$\displaystyle{\ln{{x}}}-{x}^{{{2}}}{y}={x}{y}'$$ d) $$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}+{\cos{{y}}}={\tan{{x}}}$$

Forms of linear equations

### The reduced row echelon form of the augmented matrix of a system of linear equations is given. Determine whether this system of linear equations is consistent and, if so, find its general solution. Write the solution in vector form. [1−32040\000001\000000]

Forms of linear equations

### The coefficient matrix for a system of linear differential equations of the form y′=Ay has the given eigenvalues and eigenspace bases. Find the general solution for the system. $$\displaystyleλ{1}={1}\to{\left\lbrace\begin{array}{cc} {2}&{1}\end{array}\right\rbrace},λ{2}={3}\to{\left\lbrace\begin{array}{cc} {3}&{1}\end{array}\right\rbrace}$$

Forms of linear equations

### The reduced row echelon form of the augmented matrix of a system of linear equations is given. Determine whether this system of linear equations is consistent and, if so, find its general solution. [10−6\013\000]

Vectors and spaces

### Vector Cross Product Let vectors A=(1,0,-3), B =(-2,5,1), and C =(3,1,1). Calculate the following, expressing your answers as ordered triples (three comma-separated numbers). (C) $$\displaystyle{\left({2}\overline{{B}}\right)}{\left({3}\overline{{C}}\right)}$$ (D) $$\displaystyle{\left(\overline{{B}}\right)}{\left(\overline{{C}}\right)}$$ (E) $$\displaystyle{o}{v}{e}{r}\rightarrow{A}{\left({o}{v}{e}{r}\rightarrow{B}\times{o}{v}{e}{r}\rightarrow{C}\right)}$$ (F)If $$\displaystyle\overline{{v}}_{{1}}\ \text{ and }\ \overline{{v}}_{{2}}$$ are perpendicular, $$\displaystyle{\left|\overline{{v}}_{{1}}\times\overline{{v}}_{{2}}\right|}$$ (G) If $$\displaystyle\overline{{v}}_{{1}}\ \text{ and }\ \overline{{v}}_{{2}}$$ are parallel, $$\displaystyle{\left|\overline{{v}}_{{1}}\times\overline{{v}}_{{2}}\right|}$$

Vectors and spaces

### Differentiate. $$\displaystyle{f{{\left(\theta\right)}}}=\theta{\cos{\theta}}{\sin{\theta}}$$

Vectors and spaces

### v is a set of ordered pairs (a, b) of real numbers. Sum and scalar multiplication are defined by: (a, b) + (c, d) = (a + c, b + d) k (a, b) = (kb, ka) (attention in this part) show that V is not linear space.

Vectors and spaces

### Find the scalar and vector projections of b onto a. $$\displaystyle{a}={i}+{j}+{k},{b}={i}-{j}+{k}$$

Forms of linear equations