Cross product of vectors
The cross product u×v results in a vector perpendicular to both u and v.
- Linear Combination Form: Given u=u1i+u2j+u3k and v=v1i+v2j+v3k: u×v=(u2v3−u3v2)i−(u1v3−u3v1)j+(u1v2−u2v1)k
- (x, y, z) Form / Determinant Method: u×v=detiu1v1ju2v2ku3v3
Use the cross product to find a perpendicular vector
The resulting vector from the cross product, w=u×v, is orthogonal (perpendicular) to both vector u and vector v.
Find the area of a parallelogram
The area of a parallelogram with adjacent sides defined by vectors u and v is the magnitude of their cross product.
Area=∥u×v∥
Find the image and preimage of a vector
- Image: Given a transformation T and a vector v, the image is the resulting vector T(v).
- Preimage: Given a transformation T and an image vector w, the preimage is the original vector v such that T(v)=w.
Determine if a transformation is linear
A transformation T is linear if it satisfies two conditions for all vectors u, v and any scalar c:
- Additivity: T(u+v)=T(u)+T(v)
- Homogeneity: T(cu)=cT(u)
Given a linear transformation, find the kernel and range
- Kernel: The set of all input vectors v that map to the zero vector: ker(T)={v∣T(v)=0}.
- Range: The set of all possible output vectors; the set of all images T(v). Also known as the image of T.
Given a matrix, find the kernel, nullity, range, and rank
- Kernel: The solution space of the homogeneous equation Ax=0.
- Nullity: The dimension of the kernel. It is the number of non-pivot columns (free variables) in the row echelon form of A.
- Range: The column space of A. Its basis is the set of pivot columns from the original matrix A.
- Rank: The dimension of the range (column space). It is the number of pivots in the row echelon form of A.
Find nullity given rank, or the reverse
Use the Rank-Nullity Theorem, which states that for a matrix with n columns:
rank(A)+nullity(A)=n
Find the standard matrix for a transformation
The standard matrix A for a linear transformation T is constructed by using the images of the standard basis vectors as its columns.
A=[T(e1)T(e2)…T(en)]
Once found, you can find the image of any vector v by computing T(v)=Av.
Find the standard matrix for the composition of transformations
The standard matrix of a composite transformation T=T2∘T1 (where T1 is applied first) is the product of the individual standard matrices in reverse order of application.
If A1 is the matrix for T1 and A2 is the matrix for T2, the matrix for T is:
A=A2A1
Find the coordinates of a point under a reflection, rotation, etc.
To find the coordinates of a transformed point, multiply the standard matrix for that geometric transformation (e.g., rotation, reflection) by the column vector representing the point’s coordinates.
Find the characteristic equation, eigenvalues, and eigenvectors
- Characteristic Equation: An equation used to find the eigenvalues of a square matrix A. It is given by: det(A−λI)=0
- Eigenvalues (λ): The scalar solutions to the characteristic equation.
- Eigenvectors (v): The non-zero vectors that, for a given eigenvalue λ, satisfy the equation (A−λI)v=0. A basis for the eigenspace consists of the linearly independent eigenvectors for that eigenvalue.
Find a matrix P that will diagonalize a given matrix
A matrix A is diagonalizable if it has as many linearly independent eigenvectors as it has columns. The matrix P that diagonalizes A is constructed by using these linearly independent eigenvectors as its columns.
Diagonalize a matrix
To diagonalize a matrix A, you find an invertible matrix P and a diagonal matrix D such that D=P−1AP.
- Find the eigenvalues of A.
- Find a basis of linearly independent eigenvectors for A.
- Construct the matrix P using the eigenvectors as columns.
- Construct the diagonal matrix D using the corresponding eigenvalues along the main diagonal. The order must match the order of eigenvectors in P.
Is a matrix symmetric? orthogonal?
- Symmetric: A matrix A is symmetric if it equals its transpose (A=AT).
- Orthogonal: A square matrix A is orthogonal if its columns form an orthonormal set, which means its transpose is its inverse (AT=A−1).
Find an orthogonal matrix that diagonalizes a given matrix
A matrix can be orthogonally diagonalized only if it is symmetric.
- Find the eigenvalues and corresponding eigenvectors for the symmetric matrix A.
- If any eigenspace has a dimension greater than 1, use the Gram-Schmidt process on its basis vectors to create an orthogonal basis.
- Normalize all eigenvectors by dividing each by its magnitude. This creates an orthonormal set of eigenvectors.
- Construct the orthogonal matrix P using these orthonormal eigenvectors as its columns.