Relation of null space to linear independence of columns | Summary and Q&A

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July 1, 2016
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Khan Academy
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Relation of null space to linear independence of columns

TL;DR

The video explains the relationship between the linear independence of the column vectors of a matrix and the null space of that matrix.

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Questions & Answers

Q: What is the relationship between the column vectors of a matrix and the null space of that matrix?

The column vectors of a matrix can be viewed as m-dimensional vectors. The null space of a matrix consists of all vectors that, when multiplied by the matrix, result in the zero vector.

Q: How can the linear independence of the column vectors be determined?

The column vectors of a matrix are linearly independent if and only if the only solution to the equation involving the matrix and a vector from the null space is for all components of the vector to be zero.

Q: What does it mean for a matrix to have a null space that only contains the zero vector?

If the null space of a matrix only contains the zero vector, it means that the column vectors of the matrix are linearly independent.

Q: Can a matrix have both linearly independent and linearly dependent column vectors?

No, if a matrix has linearly independent column vectors, its null space must only contain the zero vector. On the other hand, if the null space of a matrix contains only the zero vector, it means that the column vectors are linearly independent.

Summary & Key Takeaways

  • The video discusses how the column vectors of a matrix can be viewed as m-dimensional vectors.

  • The null space of a matrix is the set of all vectors that, when multiplied by the matrix, result in the zero vector.

  • The video shows that if the column vectors of a matrix are linearly independent, the null space of the matrix consists only of the zero vector.

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