23.10 - The Rank of a Matrix and Linear Independence
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Practice Questions
Test your understanding with targeted questions
Determine the rank of the following matrix: [1 2; 2 4].
💡 Hint: Look for redundancy between the rows.
List conditions when a set of vectors is considered linearly independent.
💡 Hint: Think about linear combinations and the zero vector.
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Interactive Quizzes
Quick quizzes to reinforce your learning
If a matrix has a rank equal to its number of columns, what can we say about its column vectors?
💡 Hint: Recall the definition of rank.
True or False: A set containing the zero vector can still be linearly independent.
💡 Hint: Consider the implications of the zero vector in linear combinations.
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Challenge Problems
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Consider a matrix A with vectors [(1, 2), (2, 4), (3, 6), (4, 8)]. What is its rank and what does this imply about the vectors?
💡 Hint: Check if any vector can be produced as a linear combination of the others.
If a system of equations is determined by vectors in R^6, can any matrix formed from them have a rank greater than 6? Justify your answer.
💡 Hint: Review the relationship between the rank of a matrix and its dimensional properties.
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