# Linear algebra questions and answers

Recent questions in Linear algebra
Forms of linear equations

### Determine whether the given $$(2×3)(2×3)$$ system of linear equations represents coincident planes (that is, the same plane), two parallel planes, or two planes whose intersection is a line. In the latter case, give the parametric equations for the line, that is, give equations of the form $$x=at+b , y=ct+d , z=et+f$$ $$2x_1+x_2+x_3=3$$ $$-2x_1+x_2-x_3=1$$

Alternate coordinate systems

### To fill: The blanck spaces in the statement " The origin in the rectangular coordinate system concedes with the ? in polar coordinates. The positive x-axis in rectangular coordinates coincides with the ? in polar coordinates".

Vectors and spaces

### m Find m $$Y=(6x+44)$$ $$Z=(-10x+65)$$

Vectors and spaces

### Find the distance UV between the points U(7,−4) and V(−3,−6). Round your answer to the nearest tenth, if neces

Alternate coordinate systems

### Let $$\displaystyle\beta={\left({x}^{{{2}}}-{x},{x}^{{{2}}}+{1},{x}-{1}\right)},\beta'={\left({x}^{{2}}-{2}{x}-{3},-{2}{x}^{{2}}+{5}{x}+{5},{2}{x}^{{2}}-{x}-{3}\right)}$$ be ordered bases for $$\displaystyle{P}_{{2}}{\left({C}\right)}.$$ Find the change of coordinate matrix Q that changes $$\displaystyle\beta'$$ -coordinates into $$\displaystyle\beta$$ -coordinates.

Alternate coordinate systems

### To find: The equivalent polar equation for the given rectangular-coordinate equation. Given: $$\displaystyle{x}^{2}+{y}^{2}+{8}{x}={0}$$

Matrix transformations

### Write an augumented matrix for the system of linear equations $$\displaystyle{\left[\begin{matrix}{x}-{y}+{z}={8}\\{y}-{12}{z}=-{15}\\{z}={1}\end{matrix}\right]}$$

Forms of linear equations

### The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. $$\begin{bmatrix}1&0&-1&-2&0\\0&1&2&3&0\end{bmatrix}$$

Vectors and spaces

### Determine the area under the standard normal curve that lies between ​ (a) Upper Z equals -2.03 and Upper Z equals 2.03​, ​(b) Upper Z equals -1.56 and Upper Z equals 0​, and ​(c) Upper Z equals -1.51 and Upper Z equals 0.68. ​ ​(Round to four decimal places as​ needed.)

Matrix transformations

### 1:Find the determinant of the following mattrix $$\displaystyle{\left[\begin{matrix}\begin{matrix}{2}&-{1}&-{6}\end{matrix}\\\begin{matrix}-{3}&{0}&{5}\end{matrix}\\\begin{matrix}{4}&{3}&{0}\end{matrix}\end{matrix}\right]}$$ 2: If told that matrix A is singular Matrix find the possible value(s) for x $$\displaystyle{A}={\left\lbrace\begin{matrix}{16}{x}&{4}{x}\\{x}&{9} \end{matrix}\right.}$$

Vectors and spaces

### A vector is first rotated by $$\displaystyle{90}^{\circ}$$ along x-axis and then scaled up by 5 times is equal to $$\displaystyle{\left({15},-{10},{20}\right)}$$. What was the original vector

Alternate coordinate systems

### Given point P(-2, 6, 3) and vector $$\displaystyle{B}={y}{a}_{{{x}}}+{\left({x}+{z}\right)}{a}_{{{y}}}$$, express P and B in cylindrical and spherical coordinates. Evaluate A at P in the Cartesian, cylindrical and spherical systems.

Alternate coordinate systems

### Provide notes on how triple integrals defined in cylindrical and spherical coordinates and the reason to prefer one of these coordinate systems to working in rectangular coordinates.

Forms of linear equations

### Determine whether the given $$\displaystyle{\left({2}\ \times\ {3}\right)}$$ system of linear equations represents coincident planes (that is, the same plane), two parallel planes, or two planes whose intersection is a line. In the latter case, give the parametric equations for the line, that is, give equations of the form $$\displaystyle{x}={a}{t}\ +\ {b},\ {y}={c}{t}\ +\ {d},\ {z}={e}{t}\ +\ {f}.$$ $$\displaystyle{x}_{{{1}}}\ +\ {2}{x}_{{{2}}}\ -\ {x}_{{{3}}}={2}$$ $$\displaystyle{x}_{{{1}}}\ +\ {x}_{{{2}}}\ +\ {x}_{{{3}}}={3}$$

Alternate coordinate systems

### Explain the difference between Alternating Direction Method of Multipliers(ADMM) and coordinate descent (CD) .

Vectors and spaces

### Suppose the vertices of the original figure in the example were A(-6,6), B(-2,5), and C(-6,2). What would be the vertices of the image after a 90° clockwise rotation about the origin? A'() B'() C'(___)

Vectors and spaces

### Let $$\displaystyle{B}={\left\lbrace{v}{1},{v}{2},\ldots,{v}{m}\right\rbrace}$$ be a basis for $$R^{m}$$. Suppose kvm is a linear combination of $$v1, v2, \cdots, vm-1$$ for some scalar k. What can be said about the possible value(s) of k?

Alternate coordinate systems

### (a) Find the bases and dimension for the subspace $$H = \left\{ \begin{bmatrix} 3a + 6b -c\\ 6a - 2b - 2c \\ -9a + 5b + 3c \\ -3a + b + c \end{bmatrix} ; a, b, c \in R \right\}$$ (b) Let be bases for a vector space V,and suppose (i) Find the change of coordinate matrix from B toD. (ii) Find $$\displaystyle{\left[{x}\right]}_{{D}}{f}{\quad\text{or}\quad}{x}={3}{b}_{{1}}-{2}{b}_{{2}}+{b}_{{3}}$$

Alternate coordinate systems

### (10%) In $$R^2$$, there are two sets of coordinate systems, represented by two distinct bases: $$(x_1, y_1)$$ and $$(x_2, y_2)$$. If the equations of the same ellipse represented by the two distinct bases are described as follows, respectively: $$2(x_1)^2 -4(x_1)(y_1) + 5(y_1)^2 - 36 = 0$$ and $$(x_2)^2 + 6(y_2)^2 - 36 = 0.$$ Find the transformation matrix between these two coordinate systems: $$(x_1, y_1)$$ and $$(x_2, y_2)$$.

Alternate coordinate systems