Matrix Algebra Flashcards

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.

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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.