# Recent questions in Algebra

Vectors and spaces

### Let $$\displaystyle A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$$ and 'k' be the scalar. Find the formula that relates 'detKA' to 'K' and 'detA''

Vectors and spaces

### Let the vector space $$P^{2}$$ have the inner product $$\langle p,q\rangle=\int_{-1}^{1} p(x)q(x)dx.$$ Find the following for $$p = 1\ and\ q = x^{2}.$$ $$(a) ⟨p,q⟩ (b) ∥p∥ (c) ∥q∥ (d) d(p,q)$$

Upper level algebra

### Find the x-and y-intercepts of this equation. $$\displaystyle{f{{\left({x}\right)}}}=-{x}+{2}$$

Vectors and spaces

### Find all scalars $$c_{1} , c_{2}, c_{3}$$ such that $$c_{1}(1 , -1, 0) + c_{2}(4, 5, 1) + c_{3}(0, 1, 5) = (3, 2, -19)$$

Vectors and spaces

### The position vector $$\displaystyle{r}{\left({t}\right)}={\left\langle{n}{t},\frac{{1}}{{t}^{{2}}},{t}^{{4}}\right\rangle}$$ describes the path of an object moving in space. (a) Find the velocity vector, speed, and acceleration vector of the object. (b) Evaluate the velocity vector and acceleration vector of the object at the given value of $$\displaystyle{t}=\sqrt{{3}}$$

Vectors and spaces

### A city planner wanted to place the new town library at site A. The mayor thought that it would be better at site B. What transformations were applied to the building at site A to relocate the building to site B? Did the mayor change the size or orientation of the library? [Pic]

Upper level algebra

### To calculate: The probability that the selected student was in the 37-46 are group or received a "B" in the course. Given Information: A table depicting the grade distribution for a college algebra class based on age and grade.

Matrix transformations

### Give the elementary matrix that converts [-2,-2,-1,-3,-1,-3,1,-4,-3] to [-6,-2,-1,-5,-1,-3,-7,-4,-3]

Commutative Algebra

### Let R be a commutative ring with unit element .if f(x) is a prime ideal of R[x] then show that R is an integral domain.

Upper level algebra

### A multiple regression equation to predict a student's score in College Algebra $$(\hat{y})$$ based on their high school GPA (x1x1), their high school Algebra II grade (x2x2), and their placement test score (x3x3) is given by the equation below. $$\hat{y}=-9+5x1x1+6x2x2+0.3x3x3$$ a) According to this equation, what is the predicted value of the student's College Algebra score if their high school GPA was a 3.9, their high school Algebra II grade was a 2 and their placement test score was a 40? Round to 1 decimal place. b) According to this equation, what does the student's placement test score need to be if their high school GPA was a 3.9, their high school Algebra II grade was a 2, and their predicted College Algebra score was a 67? Round to 1 decimal place.

Alternate coordinate systems

### Find the value of x or y so that the line passing through the given points has the given slope. (9, 3), (-6, 7y), $$m = 3$$

Upper level algebra

### Find the x-and y-intercepts of the graph of the equation algebraically. $$\displaystyle{\frac{{{8}{x}}}{{{3}}}}+{50}-{2}{y}={0}$$

Upper level algebra

### To find the lowest original score that will result in an A if the professor uses $$(i)(f*g)(x)\ and\ (ii)(g*f)(x)$$. Professor Harsh gave a test to his college algebra class and nobody got more than 80 points (out of 100) on the test. One problem worth 8 points had insufficient data, so nobody could solve that problem. The professor adjusted the grades for the class by a. Increasing everyone's score by 10% and b. Giving everyone 8 bonus points c. x represents the original score of a student

Upper level algebra

### To calculate: To write the given statements (a) and (b) as functions f(x) and g(x)respectively. Professor Harsh gave a test to his college algebra class and nobody got more than 80 points (out of 100) on the test. One problem worth 8 points had insufficient data, so nobody could solve that problem. The professor adjusted the grades for the class by a. Increasing everyone's score by 10% and b. Giving everyone 8 bonus points c. x represents the original score of a student

Forms of linear equations

### Write the homogeneous system of linear equations in the form AX = 0. Then verify by matrix multiplication that the given matrix X is a solution of the system for any real number $$c_1$$ $$\begin{cases}x_1+x_2+x_3+x_4=0\\-x_1+x_2-x_3+x_4=0\\ x_1+x_2-x_3-x_4=0\\3x_1+x_2+x_3-x_4=0 \end{cases}$$ $$X =\begin{pmatrix}1\\-1\\-1\\1\end{pmatrix}$$

Matrix transformations

### Show that $$\displaystyle{C}^{\ast}={R}^{\ast}+\times{T},\text{where}\ {C}^{\ast}$$ is the multiplicative group of non-zero complex numbers, T is the group of complex numbers of modulus equal to 1, $$\displaystyle{R}^{\ast}+$$ is the multiplicative group of positive real numbers.

Vectors and spaces

### Use $$\displaystyle{A}{B}←→$$ and $$\displaystyle{C}{D}←→$$ to answer the question. $$\displaystyle{A}{B}←→$$ contains the points A(2,1) and B(3,4). $$\displaystyle{C}{D}←→$$ contains the points C(−2,−1) and D(1,−2). Is $$\displaystyle{A}{B}←→$$ perpendicular to $$\displaystyle{C}{D}←→$$? Why or why not?

Matrix transformations

### Let T be the linear transformation from R2 to R2 consisting of reflection in the y-axis. Let S be the linear transformation from R2 to R2 consisting of clockwise rotation of 30◦. (b) Find the standard matrix of T , [T ]. If you are not sure what this is, see p. 216 and more generally section 3.6 of your text. Do that before you go looking for help!

Vectors and spaces