Practice Kernel And Image Of A Linear Transformation (28.4) - Linear Transformations

Practice: Kernel and Image of a Linear Transformation

Lesson 28.4Linear Transformations

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8 questions

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6 sample questions

Question 1.
Define the kernel of a linear transformation.
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Think about what happens when a transformation sends a vector to zero.
Question 2.
What is the image of a linear transformation?
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Consider all possible outputs of a transformation.
3Question 3.
What does the kernel of a linear transformation represent?
- Vector space dimension
- Set of vectors mapped to zero
- Transformations with non-trivial solutions
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Think about the significance of the zero vector.
4Question 4.
True or False: The image of a linear transformation is a subspace of the codomain.
- True
- False
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Consider how outputs behave under vector space rules.
Question 5.
Demonstrate the Rank-Nullity Theorem for a transformation T: R^3 -> R^2 defined by T(x, y, z) = (x + y, y + z).
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Compute the rank and nullity separately, then add them.
Question 6.
Given a transformation T: R^3 -> R defined by T(x,y,z) = x + y - z, find the kernel and demonstrate how this relates to the dimension of V.
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Set the transformation equation to zero and find the solution set.

4 more questions in this section. Enroll free to practise them all.