GRE Exam > GRE Videos > Mathematics for GRE Paper II > Linearly Independent & Linearly Dependent Vectors - Linear Algebra
Linearly Independent & Linearly Dependent Vectors - Linear Algebra Video Lecture | Mathematics for GRE Paper II
FAQs on Linearly Independent & Linearly Dependent Vectors - Linear Algebra Video Lecture - Mathematics for GRE Paper II
| 1. What is the definition of linearly independent vectors? |  |
Ans. Linearly independent vectors are a set of vectors that cannot be written as a linear combination of each other. In other words, if a set of vectors is linearly independent, none of the vectors in the set can be expressed as a linear combination of the others.
| 2. What is the definition of linearly dependent vectors? | |
Ans. Linearly dependent vectors are a set of vectors that can be written as a linear combination of each other. In other words, if a set of vectors is linearly dependent, at least one vector in the set can be expressed as a linear combination of the others.
| 3. How can we determine if a set of vectors is linearly independent or linearly dependent? | |
Ans. To determine if a set of vectors is linearly independent or linearly dependent, we can perform the following steps:
1. Write down the vectors as column vectors or row vectors.
2. Set up a system of equations by equating the linear combination of the vectors to the zero vector.
3. Solve the system of equations.
4. If the only solution to the system is the trivial solution (all coefficients are zero), then the vectors are linearly independent. Otherwise, they are linearly dependent.
| 4. What is the significance of linearly independent vectors in linear algebra? | |
Ans. Linearly independent vectors play a crucial role in linear algebra. They form a basis for vector spaces, which means that they can be used to represent any vector in that space. Additionally, linearly independent vectors provide a foundation for understanding concepts like vector spaces, spanning sets, and linear transformations.
| 5. Can a set of two vectors be linearly independent in three-dimensional space? | |
Ans. No, a set of two vectors cannot be linearly independent in three-dimensional space. In three-dimensional space, at least three vectors are required to form a linearly independent set. This is because a two-dimensional subspace can be spanned by two vectors, but three-dimensional space requires three linearly independent vectors to form a basis.
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