Linear Algebra - Concept Question

In summary, linear algebra is a branch of mathematics that deals with linear equations and their representations in vector spaces. Its basic concepts include vectors, matrices, linear equations, vector spaces, linear transformations, and eigenvalues and eigenvectors. It has many practical applications in fields such as computer science, engineering, economics, and physics. Learning linear algebra can improve problem-solving skills and provide a foundation for more advanced mathematical concepts. Resources for learning linear algebra include textbooks, online courses, tutorials, and various online platforms like Khan Academy and MIT OpenCourseWare.
  • #1
ChemistryNat
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Homework Statement


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The Attempt at a Solution


Since matrix B is an invertible 2x2 matrix, its row reduced echelon form will be the 2x2 identity matrix. Therefore, B, has rowspace span{[1,0][0,1]}, nullspace is the empty set and dimcol(B) is 2

Row reducing the given numerical matrix gives [(1,0),(0,1),(0,0)]
which has rowspace span{[1,0][0,1]}, nullspace is the empty set and columnspace span {[2,-1,0],[2,3,1]}Since the nullspace and rowspace of matrix B and the numerical matrix are equal, is this sufficient to say that the product, matrix A will also have the same nullspace and rowspace?

Since the numerical matrix is a 3x2 and B is 2x2, then A is 3x2
so the columnspace of A must involve column vectors with 3 variables
I'm assuming that the columns of A are linearly dependent on the columns of that explicit matrix and that the columns of A are the columns of the explicit matrix?

thank you!
 

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  • #2


Hello,

Thank you for your post. It seems like you have a good understanding of matrix B and its properties. However, to answer your question about matrix A, we need to know more information about it. The size of matrix A will depend on the operation that is being performed between matrix B and the numerical matrix.

If matrix A is the product of matrix B and the numerical matrix, then it will have the same nullspace and rowspace as matrix B. This is because the product of two invertible matrices will also be invertible, and therefore have the same nullspace and rowspace as the original matrix.

However, if matrix A is the sum or difference of matrix B and the numerical matrix, then it may have a different nullspace and rowspace. The nullspace and rowspace of a sum or difference of matrices is not necessarily the same as the individual nullspaces and rowspaces of the matrices.

In summary, if matrix A is the product of matrix B and the numerical matrix, then it will have the same nullspace and rowspace as matrix B. But if matrix A is the sum or difference of matrix B and the numerical matrix, then it may have a different nullspace and rowspace.

I hope this helps clarify things for you. Let me know if you have any further questions. Good luck with your research!
 

1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of linear equations and their representations in vector spaces. It involves the use of matrices, vectors, and linear transformations to solve problems in various fields such as physics, engineering, and economics.

2. What are the basic concepts in linear algebra?

The basic concepts in linear algebra include vectors, matrices, linear equations, vector spaces, linear transformations, and eigenvalues and eigenvectors. These concepts are used to represent and solve systems of linear equations, as well as to understand and manipulate geometric transformations.

3. How is linear algebra used in real life?

Linear algebra has many practical applications in fields such as computer science, engineering, economics, and physics. It is used to solve problems involving systems of equations, optimization, data analysis, and image processing. It is also used to model and understand complex systems such as traffic flow and neural networks.

4. What are the benefits of learning linear algebra?

Learning linear algebra can improve problem-solving skills, as it provides a systematic and logical approach to solving complex problems. It also helps in understanding and analyzing data, and provides a foundation for more advanced mathematical concepts. Additionally, many fields and industries require knowledge of linear algebra, making it a valuable skill to have.

5. What are some resources for learning linear algebra?

There are many resources available for learning linear algebra, such as textbooks, online courses, and tutorials. Some popular textbooks include "Linear Algebra Done Right" by Sheldon Axler and "Introduction to Linear Algebra" by Gilbert Strang. Online resources like Khan Academy and MIT OpenCourseWare offer free courses on linear algebra. There are also various YouTube channels and online forums dedicated to helping people learn linear algebra.

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