Composition of Linear Transformation and Matrix Multiplication2

In summary, Theorem 2.12 states that if V is an n-dimensional vector space with an ordered basis B, then the identity transformation IV is represented by the n by n matrix with "1"s down the main diagonal and "0"s everywhere else, regardless of the basis used. This is denoted by [IV]B = In. The other variables mentioned, A, B, C, D, and E, are not relevant to this theorem.
  • #1
jeff1evesque
312
0
Theorem 2.12: Let A be an mxn matrix, B and C be nxp matrices, and D and E b qxm matrices. Then
(d.) If V is an n-dimensional vector space with an ordered basis B, then [IV]B = In.

My question: What does [IV]B mean? Is this the identity matrix with respect to the vector space V which is with respect to the basis B-I'm not sure what that means. Could someone explain this in as much detail possible.

Thanks,

JL
 
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  • #2
Yes, that is exactly what it means. Specifically, it is saying that if IV(x)= x is the "identity" linear transformation on vector space V, then it is represented by the same matrix no matter what basis you use and that matrix is the n by n matrix with "1"s down then main diagonal and "0"s everywhere else, In.

(Surely, "Let A be an mxn matrix, B and C be nxp matrices, and D and E b qxm matrices" relates to something else. There is no "A", "C", "D" or "E" in what you give and B is NOT a matrix.)
 

1. What is a linear transformation?

A linear transformation is a mathematical function that maps a vector from one vector space to another, while preserving the vector operations of addition and scalar multiplication. It is also known as a linear map or linear operator.

2. What is the composition of linear transformations?

The composition of linear transformations refers to the process of applying one linear transformation after another. This results in a new linear transformation that is equivalent to performing the individual transformations in sequence.

3. How are linear transformations represented using matrices?

Linear transformations can be represented using matrices by assigning each transformation a corresponding matrix, where the columns of the matrix represent the images of the basis vectors of the input vector space.

4. How is matrix multiplication related to linear transformations?

Matrix multiplication is directly related to linear transformations, as it allows us to perform the composition of linear transformations by multiplying their corresponding matrices. This also allows us to easily apply linear transformations to vectors by multiplying them by the corresponding transformation matrix.

5. Can the composition of linear transformations be reversed?

In general, the composition of linear transformations cannot be reversed. However, there are certain cases where the composition of two linear transformations can be undone by performing the inverse of each individual transformation in reverse order.

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