Questions and Answers
What is a linear transformation?
What is the standard matrix for a linear transformation T: R^n -> R^m?
A unique matrix A such that T(x) = Ax for all x in R^n.
A mapping T: R^n -> R^m is one-to-one if each vector in R^m corresponds to at most one vector in R^n.
True
A mapping T: R^n -> R^m is onto if it maps to every vector in R^m.
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What properties must a subset H have to be considered a subspace of vector space V?
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How is the adjugate matrix formed from a square matrix A?
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What is an elementary matrix?
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What is the definition of the kernel of a linear transformation T: V -> W?
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What is the null space of an m x n matrix A?
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Define the column space of an m x n matrix A.
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What does it mean for a set of vectors to be linearly independent?
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What is a basis in a vector space V?
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What is the dimension of a vector space V?
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What is an isomorphism in linear algebra?
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Define the rank of a matrix A.
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What does the change-of-coordinates matrix do?
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What is an eigenvalue?
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Define an eigenvector.
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What is the characteristic polynomial for a matrix A?
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What does it mean for two matrices to be similar?
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What is a diagonalizable matrix?
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Define the inner product of two vectors u and v.
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What defines an orthogonal matrix?
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What is an orthonormal set?
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What is the orthogonal projection of y onto u?
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Study Notes
Linear Algebra Definitions
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Linear Transformation: A function T from vector space V to W that satisfies two main properties—additivity (T(u + v) = T(u) + T(v)) and homogeneity (T(cu) = cT(u)).
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Standard Matrix: For a linear transformation T from R^n to R^m, there is a unique matrix A that represents T such that T(x) = Ax for all x in R^n, where A = [T(e1)...T(en)].
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One-to-One Mapping: A mapping T from R^n to R^m is classified as one-to-one if each b in R^m corresponds to at most one vector x in R^n.
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Onto Mapping: A mapping T from R^n to R^m is considered onto if every vector b in R^m is the image of at least one vector x in R^n.
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Subspace: A subset H of a vector space V qualifies as a subspace if it includes the zero vector, is closed under vector addition, and is closed under scalar multiplication.
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Adjugate Matrix: Formed from a square matrix A by replacing each entry with its cofactor and transposing the result.
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Elementary Matrix: An invertible matrix created by performing a single elementary row operation on an identity matrix.
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Transpose of a Matrix: The resulting matrix from switching the rows and columns of matrix A, transforming it into dimensions n x m from m x n.
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Kernel: The kernel of a linear transformation T: V -> W consists of all vectors x in V for which T(x) equals the zero vector.
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Null Space: The collection of all solutions to the equation Ax = 0 for an m x n matrix A.
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Column Space: Represents all possible linear combinations of the columns of an m x n matrix A, denoted as Col A.
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Row Space: The set of all linear combinations of the rows of matrix A, equivalent to the column space of A transposed.
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Linear Independence: A set of vectors {v1...vp} is linearly independent if the only solution to c1v1 + ... + cpvp = 0 is when all coefficients c1, ..., cp are zero.
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Basis: A linearly independent indexed set B = {v1...vp} that spans a subspace H, indicating H = span{v1...vp}.
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Spanning Set: A collection {v1...vp} in subspace H that satisfies the condition H = span{v1...vp}.
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Dimension: Denotes the size of a basis for vector space V, represented as dim V; the dimension of the zero vector space is 0.
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Isomorphism: A linear mapping that establishes a one-to-one correspondence between two vector spaces.
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Rank: Defined as the dimension of the column space of matrix A, indicating the maximum number of linearly independent columns.
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Change-of-Coordinates Matrix: A matrix that transforms coordinate vectors from one basis B to another basis C.
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Eigenvalue: A scalar λ for which the equation Ax = λx has solutions for some nonzero vector x.
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Eigenvector: A nonzero vector x satisfying the equation Ax = λx for a given scalar λ.
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Characteristic Polynomial: For matrix A, expressed as det(A - λI), it provides the eigenvalues through its roots.
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Similar Matrices: Two matrices A and B are similar if there exists an invertible matrix P such that A = PBP^-1.
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Diagonalizable Matrix: A matrix that can be decomposed into the form PDP^-1, where D is diagonal and P is invertible.
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Inner Product: The scalar product of two vectors u and v, denoted as u^Tv or u.v, also known as the dot product.
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Orthogonal Matrix: A square matrix U where U^-1 equals U^T; such matrices feature orthonormal columns.
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Orthonormal Set: A set of vectors that are both orthogonal and unit vectors, meaning the dot product between any two different vectors in the set equals zero.
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Orthogonal Projection: The orthogonal projection of a vector y onto another vector u is calculated using the formula (y.u)/(u.u).
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