Linear Algebra: basis and matrices

In summary, a basis in linear algebra is a set of linearly independent vectors that span a vector space, while a matrix is a rectangular array used to represent linear transformations and solve systems of linear equations. To determine if a set of vectors is a basis, they must be linearly independent and span the vector space. The difference between a row vector and a column vector is in their dimensions and how they are multiplied with matrices. Linear algebra is used in various real-world applications, including image processing, computer graphics, and economics, as well as in machine learning and data analysis to model and solve complex systems.
  • #1
Shadowz
43
0
Hi,

I have a question about linear transformation. So given a matrix A in the basis u (denoted as A_u). Now in another basis that I don't know, A_u becomes A_v.

How can I find v? (I know u, A_u and A_v).

Thank you very much,
 
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  • #2
You can't in general. For instance the identity matrix has in any basis the same form:

[tex](I_u)_{ij}=(I_v)_{ij}=\delta_{ij}.[/tex]
 

1. What is a basis in linear algebra?

A basis in linear algebra is a set of linearly independent vectors that span a vector space. This means that any vector in the vector space can be written as a unique combination of basis vectors. A basis is often denoted by the symbol {v1, v2, ..., vn} where n is the number of basis vectors.

2. What is a matrix in linear algebra?

A matrix in linear algebra is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Matrices are used to represent linear transformations and can also be used to solve systems of linear equations. They are denoted by uppercase letters, such as A, B, or C.

3. How do you determine if a set of vectors is a basis?

A set of vectors is a basis if the vectors are linearly independent and span the vector space. This means that none of the vectors can be written as a linear combination of the others, and that together, they can represent any vector in the vector space. To determine if a set of vectors is a basis, you can use the row reduction method or the determinant method.

4. What is the difference between a row vector and a column vector?

A row vector is a matrix with only one row, while a column vector is a matrix with only one column. Row vectors are often used to represent points in space, while column vectors are often used to represent vectors in space. In terms of operations, row vectors are multiplied from the left by matrices, while column vectors are multiplied from the right.

5. How is linear algebra used in real-world applications?

Linear algebra is used in many real-world applications, such as image processing, computer graphics, and economics. It is also used in machine learning and data analysis to model and solve complex systems. Linear algebra is a powerful tool for solving problems with many variables and is essential in various fields, including engineering, physics, and computer science.

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