Linear Algebra Done Right: Solutions and Examples for Success

College
Algebra
Linear algebra

The Transformation of A is defined on the space $P_{2}$ of polynomials $p$ such that $\deg (p) \leq 2$ by $A p (t) = p^{'} (t)$ . Find the matrix of this transformation in the basis ${1, t, t^{2}}$ . What is $K e r (A)$ ?

vasorasy8 2022-07-05

$T (e_{1}) = T (1, 0) = (\cos θ, \sin θ)$
and
$T (e_{2}) = T (0, 1) = (- \sin θ, \cos θ)$
and
$A = [T (e_{1}) | T (e_{2})] = [\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}]$
When I rotate a vector $[\begin{matrix} x \\ y \end{matrix}]$ I get
$[\begin{matrix} x^{'} \\ y^{'} \end{matrix}] = [\begin{matrix} x \cdot \cos θ - y \cdot \sin θ \\ x \cdot \sin θ + y \cdot \cos θ \end{matrix}]$
Correct me if I'm wrong, but I thought that column 1 of A $[\begin{matrix} \cos θ \\ \sin θ \end{matrix}]$ , holds the 'x' values and column 2 holds the 'y' values. What I'm confused about is why does x' contain both an x component and a y component?

Shea Stuart 2022-07-05

Find matrix representation of transformation
Given two lines $l_{1} : y = x - 3$ and $l_{2} : x = 1$ find matrix representation of transformation $f$ (in standard base) which switch lines each others and find all invariant lines of $f$ .

malalawak44 2022-07-05

Does a non-invertible matrix transformation "really" not have an inverse?

Frank Day 2022-07-04

Suppose $T : R^{n} \to R^{m}$ is a matrix transformation. I want to show that
$T (\vec{0}) = \vec{0}$
Since $T$ is a matrix transformation, then for every $\vec{x} \in R^{n}$ there exists a unique $m \times n$ matrix $A$ such that
$T (\vec{x}) = A \cdot \vec{x}$
If we let $\vec{x} = \vec{0}$ , where this zero vector has dimensions $n \times 1$ , then
$T (\vec{0}) = A \cdot \vec{0} = \vec{0}$
where the $\vec{0}$ on the right hand side of the equation is an $m \times 1$ vector.
Hence, every transformation maps the zero vector in $R^{n}$ to the zero vector in $R^{m}$
Are there any problems with the proof?

Janessa Olson 2022-07-03

Find the equation of the line that satisfies the given conditions:
Goes through $(- 1, - 2)$ and is parallel to the line $2 x + 3 y = 8$

Joel French 2022-07-03

Find the Distance Between The Given Points:
$(\frac{1}{2}, \frac{2}{3}), (- \frac{4}{5}, \frac{1}{6})$

ntaraxq 2022-07-03

Find the equation of the line that satisfies the given conditions:
Goes through $(2, 1)$ and $(- 1, 6)$

kolutastmr 2022-07-03

Find the Distance Between The Given Points:
$(1, 1), (4, 5)$

Munawar Noori2022-07-03

A small metal bar, whose initial temperature was 20° C,
is dropped into a large container of boiling water. How
long will it take the bar to reach 90° C if it is known that
its temperature increases 2° in 1 second? How long will it
take the bar to reach 98° C

uplakanimkk 2022-07-03

Using matrix methods, how to find the image of the point (1,-2) for the transformations?
1) a dilation of factor 3 from the x-axis
2) reflection in the x-axis

prirodnogbk 2022-07-03

Consider the $m \times m$ -matrix $B$ , which is symmetric and positive definite (full rank). Now this matrix is transformed using another matrix, say $A$ , in the following manner: $A B A^{T}$ . The matrix A is $n \times m$ with $n < m$ . Furthermore the constraint $r a n k (A) < n$ is imposed.
My intuition tells me that $A B A^{T}$ must be symmetric and positive semi-definite, but what is the mathematical proof for this? (why exactly does the transformation preserve symmetry and why is it that possibly negative eigenvalues in $A$ still result in the transformation to be PSD? Or is my intuition wrong)?

delirija7z 2022-07-02

Find the Slope of the Line through P and Q
$(2, - 1), (- \frac{1}{3}, \frac{4}{5})$

pouzdrotf 2022-07-02

Find the equation of the line that satisfies the given conditions:
Goes through $(- 1, 4)$ , slope $- 9$

DIAMMIBENVERMk1 2022-07-02

Suppose $a_{k} = \frac{1}{2} p (a_{k + 1} + 1) + \frac{1}{2} p (a_{k - 1} + 1) + (1 - p) (a_{k} + 1)$ for $0 < k < n$ and $a_{0} = a_{n} = 0$ and $0 < p < 1$ . How do we find a closed form of $a_{k}$ ? For $p = 1$ , I know the solution is $a_{k} = k (n - k)$ . But I have no idea about this case. It is a system of linear equations, so I think it will a unique solution. In fact, the problem comes from random walk on ${0, 1, 2, . . ., n}$ with absorbing states $0, n$ . And $a_{k}$ is the expectation of number of steps reaching absorbing states when starting at $k$ .
$a_{k} = \frac{1}{2} p (a_{k + 1} + 1) + \frac{1}{2} p (a_{k - 1} + 1) + (1 - p) (a_{k} + 1) \Leftrightarrow p a_{k} = 1 + \frac{1}{2} p a_{k - 1} + \frac{1}{2} p a_{k + 1}$ . Thus if we let $b_{k} = p a_{k},$ , then it is equivalent to $b_{k} = 1 + \frac{1}{2} b_{k - 1} + \frac{1}{2} b_{k + 1}$ , which has solution as above. But how to solve this one about $b_{k}$ ?

mistergoneo7 2022-07-02

By conjugate linear transformation, I mean under scalar multiplication instead of $C (a f) = a C (f)$ , I would have $C (a f) = \bar{a} C (f)$ , where a is $a$ constant complex number, and $C$ is the transformation.

Nylah Hendrix 2022-07-02

An Eigenvector is nothing more than a vector that points to some place. This pointing vector will then be invariant under linear transformations.
Now my questions:
- Ok so this vector is invariant. So what? (in my case for attitude determination algorithm I even less understand what this could give me as useful information)
- how does a simple $4 \times 4$ matrix actually represent a transformation?

Wade Bullock 2022-07-02

I'm having a presentation on Gauss-Seidel iterative method, and although it isn't mandatory , I would like to have some practical examples for this method (a system of linear equations with $n \geq 1000$ , preferrably in .txt form), as well as some implementation details (maybe block GS or something, since I haven't looked that up).
I would love it if you can give me examples from real-world calculations, like in a scientific paper or something.

uri2e4g 2022-07-02

Prove that pre-multiplying a matrix $A_{m}$ by the elementary matrix obtained with any matrix elementary line transformation $I_{m} \underset{l_{1} \leftrightarrow l_{2}}{⟶} E$ is the same as applying said elementary line transformation on the matrix $A_{m}$

hryggcx 2022-07-02

Consider a linear matrix differential equation of the form
$\frac{d C}{d t} = A C + C A^{T}$
where $C$ is a symmetric $n \times n$ matrix and $A$ is a $n \times n$ matrix. Find $C (t)$ .
Is there a formal solution for the above equation? This is in principle linear equation if we treat the matrix $C$ and $A$ as a $n^{2}$ vector. However, it does not seem to be practical way to solve the problem.
This kind of differential equations for matrices is quite new to me. Besides the formal solution let me know some books considering similar topic.

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Linear algebra

Simplifying Linear Algebra: Techniques, Solutions, and Examples