Linear Algebra Definitions Flashcards

Questions and Answers

What is a linear transformation?

A process of matrix multiplication.

A method of combining multiple vectors.

A rule assigning unique vectors in W for vectors in V. (correct)

A way to measure vector lengths.

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.

True Signup and view all the answers

What properties must a subset H have to be considered a subspace of vector space V?

All of the above. Signup and view all the answers

How is the adjugate matrix formed from a square matrix A?

By replacing the (i,j) entry of A with the (i,j) cofactor and then transposing the matrix. Signup and view all the answers

What is an elementary matrix?

An invertible matrix resulting from performing one elementary row operation on an identity matrix. Signup and view all the answers

What is the definition of the kernel of a linear transformation T: V -> W?

The set of all vectors x in V such that T(x) = 0. Signup and view all the answers

What is the null space of an m x n matrix A?

The set of all solutions to Ax = 0. Signup and view all the answers

Define the column space of an m x n matrix A.

The set of all linear combinations of the columns of A. Signup and view all the answers

What does it mean for a set of vectors to be linearly independent?

There exist weights that are all equal to zero such that the linear combination results in zero. Signup and view all the answers

What is a basis in a vector space V?

An indexed set in V that is linearly independent and spans the space. Signup and view all the answers

What is the dimension of a vector space V?

The number of vectors in a basis for V. Signup and view all the answers

What is an isomorphism in linear algebra?

A one-to-one linear mapping from one vector space onto another. Signup and view all the answers

Define the rank of a matrix A.

The dimension of the column space of A. Signup and view all the answers

What does the change-of-coordinates matrix do?

Transforms B coordinate vectors into C coordinate vectors. Signup and view all the answers

What is an eigenvalue?

A scalar lambda such that Ax = lambda x for some nonzero vector x. Signup and view all the answers

Define an eigenvector.

A nonzero vector x such that Ax = lambda x for some scalar lambda. Signup and view all the answers

What is the characteristic polynomial for a matrix A?

det(A - lambda I). Signup and view all the answers

What does it mean for two matrices to be similar?

Two matrices A and B are similar if A = PBP^-1 for some invertible matrix P. Signup and view all the answers

What is a diagonalizable matrix?

A matrix that can be factored as PDP^-1, where D is a diagonal matrix and P is an invertible matrix. Signup and view all the answers

Define the inner product of two vectors u and v.

The scalar u^Tv, usually written as u.v. Signup and view all the answers

What defines an orthogonal matrix?

A square matrix U such that U^-1 = U^T. Signup and view all the answers

What is an orthonormal set?

A set of vectors that is orthogonal and consists of unit vectors. Signup and view all the answers

What is the orthogonal projection of y onto u?

The projection is given by <a href="u">(y.u)/(u.u)</a>. Signup and view all the answers

Study Notes

Linear Algebra Definitions

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)).
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)].
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.
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.
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.
Adjugate Matrix: Formed from a square matrix A by replacing each entry with its cofactor and transposing the result.
Elementary Matrix: An invertible matrix created by performing a single elementary row operation on an identity matrix.
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.
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.
Null Space: The collection of all solutions to the equation Ax = 0 for an m x n matrix A.
Column Space: Represents all possible linear combinations of the columns of an m x n matrix A, denoted as Col A.
Row Space: The set of all linear combinations of the rows of matrix A, equivalent to the column space of A transposed.
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.
Basis: A linearly independent indexed set B = {v1...vp} that spans a subspace H, indicating H = span{v1...vp}.
Spanning Set: A collection {v1...vp} in subspace H that satisfies the condition H = span{v1...vp}.
Dimension: Denotes the size of a basis for vector space V, represented as dim V; the dimension of the zero vector space is 0.
Isomorphism: A linear mapping that establishes a one-to-one correspondence between two vector spaces.
Rank: Defined as the dimension of the column space of matrix A, indicating the maximum number of linearly independent columns.
Change-of-Coordinates Matrix: A matrix that transforms coordinate vectors from one basis B to another basis C.
Eigenvalue: A scalar λ for which the equation Ax = λx has solutions for some nonzero vector x.
Eigenvector: A nonzero vector x satisfying the equation Ax = λx for a given scalar λ.
Characteristic Polynomial: For matrix A, expressed as det(A - λI), it provides the eigenvalues through its roots.
Similar Matrices: Two matrices A and B are similar if there exists an invertible matrix P such that A = PBP^-1.
Diagonalizable Matrix: A matrix that can be decomposed into the form PDP^-1, where D is diagonal and P is invertible.
Inner Product: The scalar product of two vectors u and v, denoted as u^Tv or u.v, also known as the dot product.
Orthogonal Matrix: A square matrix U where U^-1 equals U^T; such matrices feature orthonormal columns.
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.
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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Test your understanding of key linear algebra terms with these flashcards. Each card features a definition of a fundamental concept such as 'linear transformation' and 'standard matrix'. Perfect for students looking to solidify their knowledge in linear algebra.