University of Oxford mathematician Dr Tom Crawford explains the terms basis, spanning and linear independence in the context of vectors and vector spaces. Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects: [ Ссылка ]
Test your understanding of the content covered in the video with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free here: [ Ссылка ]%20-%20workbook.pdf
And here: [ Ссылка ]
You can also find several video lectures from ProPrep explaining the content covered in the video at the links below.
Basis for R^n: [ Ссылка ]/basis-for-rn/vid9962
Basis: [ Ссылка ]
Spanning: [ Ссылка ]
Linear Independence: [ Ссылка ]/linear-combination,-dependence-and-span/vid10012
As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises.
Watch the other videos from the Oxford Linear Algebra series at the links below.
Solving Systems of Linear Equations using Elementary Row Operations (ERO’s): [ Ссылка ]
Calculating the inverse of 2x2, 3x3 and 4x4 matrices: [ Ссылка ]
What is the Determinant Function: [ Ссылка ]
The Easiest Method to Calculate Determinants: [ Ссылка ]
Eigenvalues and Eigenvectors Explained: [ Ссылка ]
Spectral Theorem Proof: [ Ссылка ]
Vector Space Axioms: [ Ссылка ]
Subspace Test: [ Ссылка ]
The video begins with an intuitive example of a basis via the vector space of polynomials up to degree n. We then give the formal definition of a basis as a spanning set of linearly independent vectors.
The terms spanning and linear independence are then formally defined with examples given for each. We also show the definition of linear independence is equivalent to showing that the only solution to a linear combination of the vectors being equal to zero is for all of the coefficients to be zero. Linear dependence is defined as the lack of linear independence, or when a vector in a set can be written as a linear combination of the other vectors in the set.
Finally, we move on to a series of worked examples, beginning with several possible bases for the Cartesian plane R^2. We then look at examples of linearly independent and linearly dependent sets of vectors, and how to show this is the case. Finally, we construct two possible bases for 3D space R^3.
Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: [ Ссылка ]
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