Matrix Algebra Flashcards

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

When is a matrix in echelon form? (Select all that apply)

All entries in a column below a leading entry are zeros (correct)

All nonzero rows are above any rows of all zeros (correct)

All entries in the leading row are non-zero

Each leading entry of a row is in a column to the right of the leading entry of the row above it (correct)

What does RREF stand for?

Reduced Row Echelon Form

What is a pivot column?

A pivot position is a location that corresponds to a leading 1 in the reduced echelon form.

Define basic variable and free variable.

Basic variable: variables that correspond to pivot columns; Free variables: other variables. Signup and view all the answers

What is Span{v1...vp}?

The set of all linear combinations of v1...vp is denoted by Span {} and is called the subset of R^n spanned (or generated) by v1...vp. Signup and view all the answers

What is the definition of AX?

The product of A and x is defined as a linear combination of the columns of A using the corresponding entries of x as weights. Signup and view all the answers

What does it mean for vectors to be linearly independent?

The vector equation x1v1...xpvp = 0 has only the trivial solution. Signup and view all the answers

What is linear dependence?

There exists c1v1 + ... + cpvp = 0 for some weights c1...cp that are not all equal to 0. Signup and view all the answers

What are the conditions for a transformation to be linear?

T(U + V) = T(U) + T(V) and T(cU) = cT(U). Signup and view all the answers

When is a linear transformation T: R^n to R^m onto?

Each b in R^m is the image of at least one x in R^n. Signup and view all the answers

What is a linear transformation one-to-one?

Each b in R^m is the image of at most one x in R^n. Signup and view all the answers

State the row-column rule for computing AB.

If (AB)ij denotes the (i, j) entry in AB, then (AB)ij = ai1b1j + ai2b2j +...+ ainbnj. Signup and view all the answers

What does it mean for a linear transformation T: R^n to R^n to be invertible?

There exists function S such that S(T(x))=x and T(S(x))=x. Signup and view all the answers

What is an elementary matrix?

One that is obtained by performing a single elementary row operation on an identity matrix. Signup and view all the answers

What is the transpose of matrix A?

The transpose of A (denoted by A^T) is an nxm matrix where the columns of A are the rows of A^T. Signup and view all the answers

What is the significance of Theorem 4?

It states that for each b in R^m, Ax = b has a solution, and the columns of A span R^m. Signup and view all the answers

When does the homogeneous equation Ax = 0 have a nontrivial solution?

If the equation has at least one free variable. Signup and view all the answers

What does it mean for T to be one-to-one and onto?

T maps R^n onto R^m if and only if the columns of A span R^m; it is one-to-one if the columns of A are linearly independent. Signup and view all the answers

State the Inverse Matrix Theorem.

A is an invertible matrix if it is row equivalent to the identity matrix, has n pivot positions, the homogeneous equation Ax = 0 has only the trivial solution, and other equivalent conditions. Signup and view all the answers

If A is an invertible nXn matrix, what can be said about the solutions for each b in R^n?

There exists a unique solution x for every b in R^n. Signup and view all the answers

Study Notes

Echelon Form

A matrix is in echelon form if all nonzero rows are positioned above rows of all zeros.
Each leading entry in a row must be to the right of the leading entry in the row above.
All entries below a leading entry must be zeros.

Reduced Row Echelon Form (RREF)

Leading entries in each nonzero row are equal to 1.
Each leading 1 is the only nonzero entry in its respective column.

Pivot Columns

A column containing a leading 1 in the reduced echelon form, indicating a pivot position.

Basic and Free Variables

Basic variables correspond to pivot columns in a matrix.
Free variables are those that do not correspond to a pivot column.

Span

Span{v1...vp} represents the set of all linear combinations of vectors v1 to vp in R^n, expressed as c1v1 + c2v2 + ... + cpvp.

Product of Matrices (AX)

The product of matrix A and vector x is a linear combination of the columns of A, with corresponding entries of x serving as weights.

Linear Independence

A set of vectors is linearly independent if the equation x1v1 + ... + xpvp = 0 has only the trivial solution.
No vector in the set can be expressed as a linear combination of the others.

Linear Dependence

A set of vectors is linearly dependent if there exists a non-trivial combination that equals zero.
A zero vector in the set indicates linear dependence, as does having more variables than equations.

Linear Transformations

A transformation T is linear if it satisfies T(U+V) = T(U) + T(V) and T(cU) = cT(U).
A linear transformation from R^n to R^m is onto if every b in R^m can be produced from at least one x in R^n.
A transformation is one-to-one if it maps each b in R^m to at most one x in R^n, ensuring no free variables and linear independence of columns.

Row-Column Rule for Matrix Multiplication

If AB is defined, the entry in row i and column j is the sum of products of row i of A and column j of B.

Invertibility of a Linear Transformation

A transformation T: R^n to R^n is invertible if there exists an inverse function S such that S(T(x)) = x and T(S(x)) = x.

Elementary Matrices

Elementary matrices are obtained by performing a single elementary row operation on an identity matrix.

Transpose of a Matrix

The transpose of an mxn matrix A, denoted A^T, is an nxm matrix, where rows of A become columns of A^T.

Homogeneous Equations

The homogeneous equation Ax = 0 has a nontrivial solution if at least one free variable exists.

Linear Transformation Conditions

A linear transformation T maps R^n onto R^m if the columns of its standard matrix A span R^m.
T is one-to-one if the columns of A are linearly independent.

Inverse Matrix Theorem

A matrix A is invertible if it is row equivalent to the identity matrix, has n pivot positions, and the equation Ax = 0 has only the trivial solution.
Its columns form a linearly independent set and span R^n.

Inversion and Linear Transformations

For an invertible n x n matrix A, there is a unique solution for each b in R^n, maintaining consistency across transformations.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

Test your understanding of matrix algebra with these flashcards. Cover key concepts such as echelon form and reduced row echelon form (RREF). Perfect for anyone studying linear algebra.