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Questions and Answers
Are the vectors 6, 12, 8, -2 linearly independent?
Are the vectors 6, 12, 8, -2 linearly independent?
What does it mean if a column in the matrix is not a pivot column?
What does it mean if a column in the matrix is not a pivot column?
If a set of vectors is linearly independent, how many solutions exist to the vector equation?
If a set of vectors is linearly independent, how many solutions exist to the vector equation?
What do linearly dependent vectors imply about the constants in the vector equation?
What do linearly dependent vectors imply about the constants in the vector equation?
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How does a multiple of one vector affect linear independence?
How does a multiple of one vector affect linear independence?
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What does it mean when a set of vectors is linearly independent?
What does it mean when a set of vectors is linearly independent?
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What is the characteristic of linearly dependent vectors?
What is the characteristic of linearly dependent vectors?
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How do you determine if vectors are linearly independent?
How do you determine if vectors are linearly independent?
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When is a set of vectors considered linearly dependent?
When is a set of vectors considered linearly dependent?
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What does it imply if a set of vectors is not linearly independent?
What does it imply if a set of vectors is not linearly independent?
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In what situation would a column in the matrix be considered a pivot column?
In what situation would a column in the matrix be considered a pivot column?
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Study Notes
Linear Independence of Vectors
- A set of vectors is linearly independent if none of the vectors in the set can be written as a linear combination of the others.
- If a set of vectors is linearly independent, there is only one solution to the vector equation.
- Linear independence implies that the constants in the vector equation are all zero, except for one.
- A multiple of one vector does not affect linear independence.
Linear Dependence of Vectors
- A set of vectors is linearly dependent if one of the vectors in the set can be written as a linear combination of the others.
- Linearly dependent vectors imply that there are infinitely many solutions to the vector equation or no solution at all.
- The characteristic of linearly dependent vectors is that one vector is a linear combination of the others.
Determining Linear Independence
- To determine if vectors are linearly independent, inspect the matrix formed by the vectors as columns.
- A set of vectors is considered linearly dependent if one column is a linear combination of the others.
- A column in the matrix is considered a pivot column if it is not a linear combination of the previous columns.
Pivot Columns and Linear Independence
- If a column in the matrix is not a pivot column, it means it can be written as a linear combination of the previous columns, implying linear dependence.
- If a column in the matrix is a pivot column, it means it is not a linear combination of the previous columns, implying linear independence.
- A set of vectors is not linearly independent if one column is not a pivot column.
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Description
Test your understanding of linear independence and linear dependence of vectors. Explore how to determine if a set of vectors is linearly independent or dependent, and the concept of trivial solution in vector equations.
