Vector Subspaces and Subspace Test Explained | Linear Algebra

If V is a vector space and W is a subset of V, then W may itself be a vector space and if so is called a subspace, or linear subspace, of the vector space V. We introduce the definition of a vector subspace, the subspace test, see a proof of the subspace test, and then we'll look at several examples of subspaces and non-examples of subspaces in R^2, R^3, Mnn and more. #linearalgebra Using the Subspace Test: https://youtu.be/u_9kWr7czGI Linear Algebra course: https://www.youtube.com/playlist?list=PLztBpqftvzxWT5z53AxSqkSaWDhAeToDG Linear Algebra exercises: https://www.youtube.com/playlist?list=PLztBpqftvzxVmiiFW7KtPwBpnHNkTVeJc Join Wrath of Math to get exclusive videos, music, and more: https://www.youtube.com/channel/UCyEKvaxi8mt9FMc62MHcliw/join 0:00 Intro 0:38 Definition of a Vector Subspace 2:23 Nontrivial Subspace Properties 4:10 Subspace Test 7:38 Trivial Subspaces 9:04 Subspaces of R^2 and R^3 11:08 Non-Examples of Subspaces in R^2 and R^3 12:29 Subspaces of Square Matrices 13:50 Non-Examples of Subspaces of Square Matrices 14:27 Conclusion ◉Textbooks I Like◉ Graph Theory: https://amzn.to/3JHQtZj Real Analysis: https://amzn.to/3CMdgjI Abstract Algebra: https://amzn.to/3IjoZaO Linear Algebra: https://amzn.to/43xAWEz Calculus: https://amzn.to/3PieD1M Proofs and Set Theory: https://amzn.to/367VBXP (available for free online) Statistics: https://amzn.to/3tsaEER Discrete Math: https://amzn.to/3qfhoUn Number Theory: https://amzn.to/3JqpOQd ★DONATE★ ◆ Support Wrath of Math on Patreon for early access to new videos and other exclusive benefits: https://www.patreon.com/join/wrathofmathlessons ◆ Donate on PayPal: https://www.paypal.me/wrathofmath Thanks to Loke Tan, Raül Beienheimer, Matt Venia, Micheline, Doug Walker, Odd Hultberg, Marc, Shlome Ashkenazi, Barbora Sharrock, Mohamad Nossier, Rolf Waefler, Shadow Master, and James Mead for their generous support on Patreon! Outro music is mine. You cannot find it anywhere, for now. Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● Facebook: https://www.facebook.com/WrathofMath ● Twitter: https://twitter.com/wrathofmathedu My Math Rap channel: https://www.youtube.com/channel/UCQ2UBhg5nwWCL2aPC7_IpDQ/featured