21.14.4 - Rank-Nullity Theorem
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Practice Questions
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Define the kernel of a transformation.
💡 Hint: Consider what happens to vectors during transformation.
What is the rank of a linear transformation?
💡 Hint: Think about the dimensional space that the transformation maps to.
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Interactive Quizzes
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What does the Rank-Nullity Theorem establish?
💡 Hint: Remember how rank and nullity relate to each other.
True or False: The dimension of the kernel is equal to the dimension of the image.
💡 Hint: Think about how different dimensions affect transformations.
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Challenge Problems
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Given a transformation T from R^5 to R^2, where Ker(T) has dimension 3. Determine the rank and comment on the implications for engineering systems.
💡 Hint: Recall how the dimensions interplay via the Rank-Nullity Theorem.
A civil engineering model uses a transformation T: R^6 -> R^3, with a null space of dimension 4. What is the rank? How does this apply to model predictions?
💡 Hint: Apply the Rank-Nullity equation to derive ranks and summaries.
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