Tag Archives: Linear Algebra

Superposition Principle of Linear Algebra

Superposition Principle If is a solution to , and is a solution to , then is a solution to . Proof: Properties of Matrix Transpose 1) 2) 3) 4)

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Characteristic Polynomial, Eigenvalues, Eigenvectors

Characteristic Polynomial, Eigenspace The solution space of is called the eigenspace of associated with the eigenvalue . The eigenspace is denoted by . Sum/Product of Eigenvalues – The sum of all eigenvalues of (including repeated eigenvalues) is the same as … Continue reading

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Finding Least Squares Solution Review and Others

Rotation Matrix The rotation matrix rotates points in the -plane counterclockwise through an angle about the origin. For example rotating the vector 45 degrees counterclockwise gives us: Finding Least Squares Solution Given (inconsistent system), solve instead to get a least … Continue reading

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Dot Product and Span Summary

Dot Product – – Span Subspaces is a subspace of if 1) for some vectors . 2) satisfies the closure properties: (i) for all , we must have . (ii) for all and , we must have . 3) is … Continue reading

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Gaussian Elimination Summary

Row echelon form (REF) For each non-zero row, the leading entry is to the right of the leading entry of the row above. E.g. Note that the leading entry 9 of the second row is to the right of the … Continue reading

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Vector Subspace Question (GRE 0568 Q3)

This is an interesting question on vector subspaces (a topic from linear algebra): Question: If V and W are 2-dimensional subspaces of , what are the possible dimensions of the subspace ? (A) 1 only (B) 2 only (C) 0 … Continue reading

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Thanks for reading our blog Mathtuition88.com. We are also on: Facebook: https://www.facebook.com/mathtuition88 Twitter: https://twitter.com/mathtuition88 Google+: Mathtuition88 If you are a subscriber, thanks for your support! We will continue our mission of blogging interesting articles on Mathematics! Featured book: Coding the Matrix: Linear … Continue reading

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3×3 Matrix Inverse Shortcut (Fastest Method)

Source: http://www.math.columbia.edu/~bayer/LinearAlgebra/pdf/inverse.pdf Backup copy (In case of broken link): 3×3 Matrix Inverse This is an excellent method for finding the inverse of a 3×3 matrix, probably the fastest and easiest method for 3×3. (Row reduction is better for 4×4 matrices and above.) … Continue reading

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Linear Algebra Resources

1) https://www.math.okstate.edu/~binegar/3013-S99/3013-l16.pdf (How to diagonalize matrices) 2) http://www.sosmath.com/matrix/expo/expo.html (How to find exponential of matrices) 3)  http://en.wikipedia.org/wiki/Matrix_differential_equation (Matrix Differential Equations) 4) http://www.ucl.ac.uk/CNDA/courses/coursework/JNF.pdf (Jordan Normal Form) 5) http://www.math.niu.edu/~fletcher/jnf.pdf (Jordan Normal Form for 2×2 Matrices) 6) http://www.math.colostate.edu/~gerhard/M345/CHP/ch9_6.pdf (Matrix Exponential with repeated Eigenvalues) 7) http://www.math.vt.edu/people/afkhamis/class_home/notes/F08W12.pdf http://www-users.math.umn.edu/~leif0020/Spring2011/matrixexp.pdf (Matrix Differential equation with … Continue reading

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