Oxford Linear Algebra: Dimension Formula for Vector Spaces

Описание: University of Oxford mathematician Dr Tom Crawford introduces the dimension formula for vector spaces via a worked example before going through a complete proof. Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects: https://www.proprep.uk/info/TOM-Crawford

Links to the other videos mentioned:
Subspace Test -    • Oxford Linear Algebra: Subspace Test  
Basis, Spanning and Linear Independence -    • Oxford Linear Algebra: Basis, Spanning and...  

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: https://api.proprep.com/course/downlo...

You can also find several video lectures from ProPrep explaining the content covered in the video at the links below.

Dimension Formula: https://www.proprep.com/university-of...

Sum of Subspaces: https://www.proprep.com/university-of...

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):    • Oxford Linear Algebra: Elementary Row Oper...  

Calculating the inverse of 2x2, 3x3 and 4x4 matrices:    • Oxford Linear Algebra: How to find a Matri...  

What is the Determinant Function:    • Oxford Linear Algebra: What is the Determi...  

The Easiest Method to Calculate Determinants:    • Oxford Linear Algebra: The Easiest Method ...  

Eigenvalues and Eigenvectors Explained:    • Oxford Linear Algebra: Eigenvalues and Eig...  

Spectral Theorem Proof:    • Oxford Linear Algebra: Spectral Theorem Proof  

Vector Space Axioms:    • Oxford Linear Algebra: What is a Vector Sp...  

Subspace Test:    • Oxford Linear Algebra: Subspace Test  

Basis, Spanning and Linear Independence:    • Oxford Linear Algebra: Basis, Spanning and...  

The video begins by defining the dimension of a vector space as the number of elements in its basis. This is then exemplified by looking at the vector space of polynomials up to degree n, which has a dimension of n+1.

We then go through a fully worked example in R^4 by calculating explicitly the dimensions of the subspaces X and Y, the subspace X+Y, and the intersection of X and Y. This is used as motivation for the dimension formula: dim(X+Y) = dim(X) + dim(Y) - dim(X and Y).

Finally, a complete proof of the dimension formula is presented where we construct a basis of the space X+Y which is shown to be both spanning and linearly independent.

Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: https://www.seh.ox.ac.uk/people/tom-c...

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