🤖 GPT

🧠Unleash your full potential and become an expert in your field with this amazing prompt!🧠
This prompt will make complex concepts understandable to everyone!
Get a personalized, detailed and comprehensive curriculum for mastering the subject, complete with practical examples and step-by-step guides.
This prompt uses role play and educator structure that allows you to become trained by the top university professors in any academic study!

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$5.99

Model

Chat - **GPT-3.5 **(gpt-3.5-turbo)

Token size

198 ($0.00040 / call)

Example input

First blank: mathematic
Second blank: linear algebra
If you understand this respond with "Nice to meet you!", and start.

Example output

Professor:
Nice to meet you! As an Ivy League mathematics professor, I am excited to help you learn linear algebra. This course will cover the fundamentals of linear algebra, including vectors, matrices, linear transformations, and eigenvectors and eigenvalues.
Here is a comprehensive syllabus for the course:
I. Vectors in Linear Algebra
A. Introduction to Vectors
What is a vector?
Vector addition and scalar multiplication
Vector algebra and geometry
B. Linear Combinations and Span
Linear combinations of vectors
Span of a set of vectors
Linear independence and bases
II. Matrices and Linear Transformations
A. Introduction to Matrices
What is a matrix?
Matrix addition and scalar multiplication
Matrix algebra and geometry
B. Matrix Multiplication and Inverse
Matrix multiplication and properties
Inverse of a matrix
C. Linear Transformations
What is a linear transformation?
Matrix representation of linear transformations
Composition of linear transformations
III. Eigenvectors and Eigenvalues
A. Introduction to Eigenvectors and Eigenvalues
What are eigenvectors and eigenvalues?
Computing eigenvectors and eigenvalues
B. Applications of Eigenvectors and Eigenvalues
Diagonalization of matrices
Solving linear differential equations
Applications in physics and engineering
At the end of each section, please let me know if you need any more explanation or examples, or if you are ready to move on to the next section. If you are ready to proceed, we will start with Section I: Vectors in Linear Algebra.
User:
lets start
Professor:
Great! Let's begin with Section I: Vectors in Linear Algebra.
A. Introduction to Vectors
What is a vector?
A vector is a mathematical object that has both magnitude and direction. In linear algebra, we typically represent vectors as column matrices, where each entry in the matrix corresponds to a component of the vector.
For example, consider the vector v = (2, 3). We can represent this vector as the following column matrix:
2
3
Vector addition and scalar multiplication
We can add vectors of the same dimension by adding their corresponding components. For example, if we have two vectors u = (1, 2) and v = (3, 4), then their sum is given by u + v = (1 + 3, 2 + 4) = (4, 6).
We can also multiply a vector by a scalar, which multiplies each component of the vector by the scalar. For example, if we have the vector v = (2, 3) and the scalar c = 2, then the scalar multiple of v by c is given by cv = (2c, 3c) = (4, 6).
Vector algebra and geometry
Vectors can be used to represent geometric objects, such as points, lines, and planes. For example, a line passing through the point (1, 2) with direction vector (3, 4) can be represented parametrically as the set of all points of the form (1 + 3t, 2 + 4t), where t is a real number.
B. Linear Combinations and Span
Linear combinations of vectors
A linear combination of vectors is a sum of scalar multiples of those vectors. For example, if we have vectors v1 = (1, 2) and v2 = (3, 4), then a linear combination of v1 and v2 is any vector of the form c1v1 + c2v2, where c1 and c2 are scalars.
Span of a set of vectors
The span of a set of vectors is the set of all linear combinations of those vectors. For example, the span of the vectors v1 = (1, 2) and v2 = (3, 4) is the set of all vectors of the form c1(1, 2) + c2(3, 4), where c1 and c2 are scalars.
Linear independence and bases
A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the other vectors in the set. A basis for a vector space is a set of linearly independent vectors that span that space.
For example, the vectors v1 = (1, 0) and v2 = (0, 1) form a basis for the two-dimensional vector space R^2.
Now that you have learned about vectors, linear combinations, and span, are there any concepts or examples you would like me to clarify further, or are you ready to move on to the next section?

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