Linear Algebra Done Right: Solutions and Examples for Success

College
Algebra
Linear algebra

Which transformations include in the matrix?
do this matrix transformation options:
1) rotate and scaling
2) translate and scaling
3) scaling and rotate
4) not 1,2,3
$[\begin{matrix} 1 & 0 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 1 \end{matrix}]$

sg101cp6vv 2022-05-15

Let $T : R^{2} \to R^{3}$ and $T (- 2, 3) = (- 1, 0, 1)$ and $T (1, 2) = (0, - 1, 0)$
Obtain the canonical matrix of $T$ and the transformation $T (x, y)$ .

Jaylene Duarte 2022-05-15

Let $Σ$ be a symmetric positive definite matrix with ones on the diagonal (= correlation matrix).
Let $A$ be an invertible matrix.
I'm pretty sure that if $Ω := A Σ A^{t}$ has ones on its diagonal, then
$Ω = I d or Ω = Σ$
(which would correspond to $A = I d$ or $A = c h o l (Σ)$ )
But I don't know how to proove it.

fetsBedscurce4why1 2022-05-14

Let $B = {u_{1}, u_{2}, u_{3}}$ with $u_{1} = 1$ , $u_{2} = x$ , $u_{3} = x^{2}$ denote a basis for the space of polynomials of the second order polynomials $P^{2}$ . Let $T (a_{0} + a_{1} x + a_{2} x^{2}) = a_{0} + a_{1} (x - 1) + a_{2} (x - 1)^{2}$ be a linear transformation form $P^{2}$ to itself. Construct the representation matrix $[T]_{B B} .$ .

ureji1c8r1 2022-05-13

We know that if we want to reflect any point over an origin, i.e. $O (0, 0)$ , we can use matrix transformation like this
$(\begin{matrix} x^{'} \\ y^{'} \end{matrix}) = (\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} - x \\ - y \end{matrix}) .$
But, what if we reflect any point over another point $M (a, b)$ with $a, b \neq 0$

kwisangqaquqw3 2022-05-13

How to formulate a transformation matrix for the following operation? Like all the examples I found are different and I can't understand how to solve this problem:
$y = A . X y = (y_{1} y_{2})^{'} x = (x_{1}^{'} x_{2}) (y_{1} y_{2})^{'} = (x_{1} x_{2})^{'} (- x_{1}, x_{2})^{'} = (- x_{1}, - x_{2})^{'}$

Deshawn Cabrera 2022-05-13

following matrix
$[\begin{matrix} 2 & 0 \\ 0 & 2 \end{matrix}]$
Results in the shape being twice as small.
Now I am trying to use this matrix and apply it somewhere else, but because of the inverse scale I am unable to use it. Is there any way to normalize the scaling factor of a affine transformation matrix?
One last thing to note is the fact that the rotation applied, according to documentation is also reversed from standard formulas. Their rotation matrix would look something like this:
$[\begin{matrix} c o s & s i n \\ - s i n & c o s \end{matrix}]$

Two 3 dimensional points. $A [x_{1}, y_{1}, z_{1}]$ and $B [x_{2}, y_{2}, z_{2}]$ . Need to find a transformation matrix which when multiplied to A will give me B and when multiplied by B give me A. The transformation needs to be a reflection against the plane that's perpendicular to the middle of the AB segment and passing through the midpoint of the AB.

lifretatox8n 2022-05-12

In the vectorspace $R^{2}$ two vectors are given (Which also form the basis a): $a_{1} = (8, - 3)$ and $a_{2} = (5, - 2)$
A linear mapping is determined by: $f (a_{1}) = 2 a_{1} - 4 a_{2}$ and $f (a_{2}) = - a_{1} + 2 a_{2}$
How can one determine the transformation matrix of f with respect to the standard e-basis?

In each part suppose that the augmented matrix for a system of linear equation has been reduced by row operations to the given row echelon form. Solve the system

Irum Tariq2022-05-08

Suppose that the augmented matrix of a linear equations has been partially reduced using elementary row operations to

1 1 3 5
0 1 1 r
0 s 1 2
Find all the value(s) (if any) of r and s for which the given system has:
(a) a unique solution, (b) an infinitely many solutions, (c) No solutions.

Godsway Attuh-Ndro2022-05-08

What is the difference between solving an equation such as 5y + 3 - 4y - 8 = 6 + 9 and simplifying an algebraic expression such as 5y + 3 - 4y - 8 ? If there is a difference, which topic should be taught first ? Why ?

Suppose $U = {(x, y, x + y, x - y, 2 x) \in F^{5}, x, y \in F}$ . Find a subspace $W$ of $F^{5}$ such that $F^{5} = U \oplus W$ .

Teresa Sanchez2022-04-25

For the displacement of the element by n rows from the identity matrix, the element should hold the value $(\frac{1}{2})^{n}$ in the transformed matrix.
Here are a few examples:
$A = (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) \to A^{'} = (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix})$
$B = (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}) \to B^{'} = (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & \frac{1}{2} \\ 0 & \frac{1}{2} & 0 \end{matrix})$
$C = (\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix}) \to C^{'} = (\begin{matrix} 0 & \frac{1}{2} & 0 \\ 0 & 0 & \frac{1}{2} \\ \frac{1}{4} & 0 & 0 \end{matrix})$
How can such a transformation be realized mathematically?

Tristan Meyers 2022-04-12

Consider a triangle that has vertices at A(1,2), B(2,3) and C(4,2). Reflect this triangle in the line through the origin which is inclined at $30^{\circ}$ to the positive x-axis. Find the vertices of the transformed triangle.
To solve this question, I rotate this triangle through $- 30^{\circ}$ first and then reflect it about x-axis and then rotate through $30^{\circ}$ . I am not sure if this way is correct.

affizupdaftf3opg 2022-04-12

How would one go about analytically solving a system of non-linear equations of the form:
$a + b + c = 4$
$a^{2} + b^{2} + c^{2} = 6$
$a^{3} + b^{3} + c^{3} = 10$
Thank you so much!

Finding detailed linear algebra problems and solutions has always been difficult because the textbooks would never provide anything that would be sufficient. Since it is used not only by engineering students but by anyone who has to work with specific calculations, we have provided you with a plethora of questions and answers in their original form. It will help you to see some logic as you are solving complex numbers and understand the basic concepts of linear Algebra in a clearer way. If you need additional help or would like to connect several solutions, compare more than one solution as you approach your task.

Linear algebra

Which transformations include in the matrix?
do this matrix transformation options:
1) rotate and scaling
2) translate and scaling
3) scaling and rotate
4) not 1,2,3
$[\begin{matrix} 1 & 0 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 1 \end{matrix}]$

Let $T : R^{2} \to R^{3}$ and $T (- 2, 3) = (- 1, 0, 1)$ and $T (1, 2) = (0, - 1, 0)$
Obtain the canonical matrix of $T$ and the transformation $T (x, y)$ .

$y \leq 4$

In each part suppose that the augmented matrix for a system of linear equation has been reduced by row operations to the given row echelon form. Solve the system

Suppose that the augmented matrix of a linear equations has been partially reduced using elementary row operations to

1 1 3 5
0 1 1 r
0 s 1 2
Find all the value(s) (if any) of r and s for which the given system has:
(a) a unique solution, (b) an infinitely many solutions, (c) No solutions.

What is the difference between solving an equation such as 5y + 3 - 4y - 8 = 6 + 9 and simplifying an algebraic expression such as 5y + 3 - 4y - 8 ? If there is a difference, which topic should be taught first ? Why ?

a−15=4a−3

Suppose $U = {(x, y, x + y, x - y, 2 x) \in F^{5}, x, y \in F}$ . Find a subspace $W$ of $F^{5}$ such that $F^{5} = U \oplus W$ .

[ k 1 −2 ,4 −1 2]

How would one go about analytically solving a system of non-linear equations of the form:
$a + b + c = 4$
$a^{2} + b^{2} + c^{2} = 6$
$a^{3} + b^{3} + c^{3} = 10$
Thank you so much!

Simplifying Linear Algebra: Techniques, Solutions, and Examples