University of Oxford mathematician Dr Tom Crawford explains the subspace test for 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%20uk.pdf

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You can also find several video lectures from ProPrep explaining subspaces here: [ Ссылка ]/subspaces/vid9982/

And further videos explaining subspaces for more general vector spaces here: [ Ссылка ]

As with all modules on ProPrep, each set of videos contains lectures, worked examples and full solutions to all exercises.

Watch 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 ]

The video begins with the definition of a subspace U contained in a vector space V, and some trivial examples for U = V and U = 0. The subspace test is then introduced and shown to be equivalent to the definition. The subspace test requires the zero vector to be contained in U, and any linear combination of vectors in U to also be contained in U. Finally, 3 fully worked examples are shown. First, we show that the x-y plane is a subspace of 3-dimensional coordinate space. Second, we show that for U and W subspaces of a vector space V, the intersection of U and W is always a subspace. Third, we show that the subspace of differentiable functions from the real numbers to the real numbers is a subspace of the vector space of all functions from R to R.

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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