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Linear transformations Video Lecture | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
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FAQs on Linear transformations Video Lecture - Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
| 1. What is a linear transformation? |  |
A linear transformation is a mathematical function that maps vectors from one vector space to another, while preserving the straightness of lines and the origin. It can be represented by a matrix multiplication, where each element of the vector is multiplied by a corresponding element in the matrix and then summed up to obtain the transformed vector.
| 2. What are the properties of a linear transformation? | |
There are three main properties of a linear transformation:
1) Preservation of addition: For any two vectors u and v in the vector space, the linear transformation T(u+v) is equal to T(u) + T(v).
2) Preservation of scalar multiplication: For any scalar c and vector v, the linear transformation T(cv) is equal to cT(v).
3) Preservation of the origin: The linear transformation T(0) is equal to 0, where 0 represents the zero vector in the vector space.
| 3. How can we represent a linear transformation using matrices? | |
A linear transformation can be represented using matrices by defining a matrix that corresponds to the transformation. The columns of the matrix are formed by applying the transformation to the basis vectors of the vector space. When we multiply this matrix with a vector, we obtain the transformed vector.
| 4. What is the relationship between linear transformations and eigenvalues/eigenvectors? | |
Eigenvalues and eigenvectors are closely related to linear transformations. An eigenvector of a linear transformation is a non-zero vector that only gets scaled by the transformation, without changing its direction. The corresponding eigenvalue is the scalar value by which the eigenvector is scaled. Eigenvectors/eigenvalues provide important information about the behavior of linear transformations, such as their stretch or compression along certain directions.
| 5. Can all transformations be represented as linear transformations? | |
No, not all transformations can be represented as linear transformations. Linear transformations have specific properties, such as preserving addition and scalar multiplication, which many other transformations do not satisfy. Non-linear transformations, for example, may involve curves, rotations, or any other alteration of the shape or structure of the vectors being transformed.
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Dec 22, 2024 Last updated |
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