[Linear Algebra] Matrix Transformations

In summary, the person had problems with matrix transformations and is looking for help. They found two problems and one solution. The first problem was to map from a standard basis to a B basis and the second problem was to solve for P in an equation. The solution for the first problem was to use left multiplication and the solution for the second problem was to use left multiplication and the transformation matrix from the B basis to the standard basis.
  • #1
Homework Statement
https://prnt.sc/riq7m0
Relevant Equations
Gauss Jordan
Evening,

The reason for this post is because as the title suggests, I have a question concerning matrix transformation. These are essentially test prep problems and I am quite stuck to be honest.

Here are the [questions](https://prnt.sc/riq7m0) and here are the [answers](https://prnt.sc/riq8b6).

For problem **#14**, pretty simple. [Here](https://prnt.sc/riqb0v) is my solution.

Now, for problem **#15a**, I'm confused. The solution states and solves for P^-1 . How and why? Shouldn't #14 and #15a share the same answer of P given i am simply asked to find the transition matrix from the basis B to the basis B'? And even then, i didn't find the given answer to be the inverse of P so how exactly can i go about solving this.?

As for problem **#15b**, i tried doing the reverse of problem #14 as i now am asked to solve BB' rather than B'B. So, in solving BB', i got an answer that doesn't resemble the given answer in the slightest. Not sure what I'm doing wrong.

Seems i have my whole concept of how to go about transformations misconstrued. Please help. Any comments/suggestions would be greatly appreciated. Thank you in advance and good day. Cheers!)
 
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  • #2
Consider how you would map the standard basis [itex]\left\{ \left[\begin{array}{c} 1\\0 \end{array}\right], \left[\begin{array}{c} 0\\1 \end{array}\right]\right\}[/itex]to either B or B' and then back again.
Then note that these transformations can be composed by multiplying the transformation matrices.
So Transform From B to Standard Basis and then to B' (and then the reverse).

Secondly note that left multiplication by a square matrix transforms column vectors and columns of a general matrix in the same way. So you can transform both basis vectors by forming their columns into a 2x2 matrix and left multiplying by your transformation matrix. [itex] T\cdot B_1 = B_2 [/itex] as a product of two square matrices forming a square matrix.
 
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1. What is a matrix transformation?

A matrix transformation is a mathematical operation that maps one set of values to another set of values using a matrix. This transformation can change the size, shape, or orientation of the original set of values, and is often used to solve systems of linear equations.

2. How is a matrix transformation represented?

A matrix transformation is typically represented using a matrix notation, where the original set of values is written as a vector and the transformed set of values is written as a new vector multiplied by a transformation matrix. The transformation matrix contains the coefficients that determine how the values are transformed.

3. What are the different types of matrix transformations?

There are several types of matrix transformations, including translation, scaling, rotation, reflection, and shearing. These transformations can be applied in any combination and order to achieve different effects on the original set of values.

4. How are matrix transformations used in real life?

Matrix transformations have many practical applications in various fields, including computer graphics, engineering, physics, and economics. They are used to model and solve complex systems, such as predicting the behavior of a physical system or analyzing financial data.

5. What are some properties of matrix transformations?

Matrix transformations have several important properties, including linearity, invertibility, and associativity. Linearity means that the transformation preserves the properties of a straight line, while invertibility means that the transformation can be reversed. Associativity means that the order in which multiple transformations are applied does not affect the final result.

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