Practice Linear Independence And Dependence (26.5) - Vector Spaces - Mathematics (Civil Engineering -1)

Practice: Linear Independence and Dependence

Lesson 26.5Vector Spaces

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8 questions

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6 sample questions

Question 1.
Determine if the vectors v₁ = (1, 2) and v₂ = (-2, -4) are linearly independent.
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Look for a scalar multiple.
Question 2.
Are the vectors v₁ = (0, 1) and v₂ = (1, 0) independent?
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Think about their directions.
3Question 3.
What does it mean for a set of vectors to be linearly independent?
- They can be expressed in terms of one another
- They cannot express the zero vector unless all coefficients are zero
- They always lie on the same line
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Think about how combinations of coefficients work.
4Question 4.
True or False: If one vector in a set can be expressed as a linear combination of the others, the set is linearly independent.
- True
- False
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Review the definitions of dependence and independence.
Question 5.
Given the vectors (1,2,3), (2,4,6), and (3,6,9), demonstrate whether they are linearly independent or dependent. Prove your answer mathematically.
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Get the system of equations formed by their linear combination equating to zero.
Question 6.
Take the vectors (1,0,0), (0,1,0), and (0,0,1). Show they span R³ and are independent.
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Try forming the identity matrix with these as columns.

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