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Linear Transformation & Examples of Linear Transformation - Linear Algebra Video Lecture | Engineering Mathematics - Civil Engineering (CE)
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FAQs on Linear Transformation & Examples of Linear Transformation - Linear Algebra Video Lecture - Engineering Mathematics - Civil Engineering (CE)
| 1. What is a linear transformation? |  |
A linear transformation, also known as a linear map, is a function between two vector spaces that preserves the operations of addition and scalar multiplication. It maps vectors from one space to another in a linear manner, meaning that it maintains the linearity property: T(a*u + b*v) = a*T(u) + b*T(v), where T represents the linear transformation, u and v are vectors, and a and b are scalars.
| 2. What are some examples of linear transformations? | |
Some examples of linear transformations include:
- Scaling: The transformation that stretches or shrinks a vector by a constant factor in each dimension.
- Rotation: The transformation that rotates a vector around a fixed point or axis.
- Reflection: The transformation that mirrors a vector across a line or plane.
- Projection: The transformation that projects a vector onto a lower-dimensional subspace.
- Translation: The transformation that shifts a vector by a fixed amount in each dimension.
| 3. How can I determine if a transformation is linear? | |
To determine if a transformation is linear, you need to check if it preserves the linearity property. This means that the transformation should satisfy the following two conditions:
1. T(u + v) = T(u) + T(v) for any vectors u and v.
2. T(c*u) = c*T(u) for any scalar c and vector u.
If both of these conditions hold true for a given transformation, then it is linear. However, if either condition fails for any pair of vectors or scalars, the transformation is not linear.
| 4. Can a linear transformation change the dimension of a vector space? | |
No, a linear transformation cannot change the dimension of a vector space. The dimension of the vector space is determined by the number of linearly independent vectors needed to span the space. Since a linear transformation preserves the operations of addition and scalar multiplication, it cannot introduce new linearly independent vectors or remove existing ones. Therefore, the dimension of the vector space remains the same after a linear transformation.
| 5. How are linear transformations represented mathematically? | |
Linear transformations can be represented mathematically using matrices. For example, consider a linear transformation T that maps vectors from a vector space V to another vector space W. If {v1, v2, ..., vn} is a basis for V and {w1, w2, ..., wm} is a basis for W, then T can be represented by an m x n matrix A. Each column of the matrix corresponds to the image of the corresponding basis vector in V under the transformation T. The result of applying the transformation to a vector v can be obtained by multiplying the matrix A with the column vector representation of v.
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Dec 22, 2024 Last updated |
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