Linear Independence
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Mathematics
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11th Grade - University
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Practice Problem
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Hard
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10 questions
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1.
MULTIPLE CHOICE QUESTION
30 sec ⢠1 pt
What does it mean for vectors to be linearly dependent?
They are always orthogonal.
They have the same magnitude.
They cannot be expressed as a linear combination of each other.
They can be expressed as a linear combination of each other.
2.
MULTIPLE CHOICE QUESTION
30 sec ⢠1 pt
In a vector space, when are vectors considered linearly independent?
When the vectors have different magnitudes.
When the vectors have the same direction.
When the only solution to the linear combination is all scalars being zero.
When the sum of the vectors is zero.
3.
MULTIPLE CHOICE QUESTION
30 sec ⢠1 pt
What is the result of solving a system of equations if the vectors are linearly independent?
Multiple solutions with free variables.
A unique solution with all scalars being zero.
A set of nonzero solutions.
No solution at all.
4.
MULTIPLE CHOICE QUESTION
30 sec ⢠1 pt
What is a free variable in the context of linear independence?
A variable that is always positive.
A variable that can only be zero.
A variable that is always negative.
A variable that can take any value.
5.
MULTIPLE CHOICE QUESTION
30 sec ⢠1 pt
How can row reduction help in determining linear independence?
By calculating the determinant.
By transposing the matrix.
By simplifying the matrix to row echelon form.
By finding the inverse of the matrix.
6.
MULTIPLE CHOICE QUESTION
30 sec ⢠1 pt
What does a zero determinant indicate about a set of vectors?
The vectors have equal magnitudes.
The vectors are linearly independent.
The vectors are linearly dependent.
The vectors are orthogonal.
7.
MULTIPLE CHOICE QUESTION
30 sec ⢠1 pt
What is the significance of a nonzero determinant in a square matrix?
It indicates the matrix is symmetric.
It indicates the matrix is singular.
It indicates linear independence.
It indicates linear dependence.
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