Linear Algebra Flashcards

What does 'A' represent?

What is 'b' in relation to the identity matrix?

Does 'c' mean that A has more than n pivots?

Does 'd' signify that ax=0 has only non-trivial solutions?

Does 'e' imply that the columns of A form a linearly independent set?

Does 'f' mean that the transformation x -> Ax is not one to one?

Does 'g' signify that the equation ax=b has at least one solution for every b in R^n?

Is 'h' stating that the columns of A do not span R^n?

Does 'i' describe the transformation x -> ax mapping R^n onto R^n?

Does 'j' mean there does not exist an nxn matrix c such that CA = I?

Does 'k' indicate the existence of an nxn matrix d such that AD = I?

Is 'l' stating that A^t is not invertible?

Does 'm' mean that the columns of A do not form a basis of R^n?

Does 'n' state that Col A is not equal to R^n?

Is 'o' referring to a dimension of Col A that is not equal to n?

Does 'p' imply that the rank of A is less than n?

Is 'q' stating that Nul A is not equal to {0}?

Does 'r' imply that dim Nul A is greater than 0?

Invertible Matrix Properties

A matrix A is defined as invertible when there exists another matrix, denoted as C, such that the product CA equals the identity matrix I.
The matrix must also be row equivalent to the n x n identity matrix, indicating full rank.

Pivots and Solutions

An invertible matrix A contains n pivots, which are the leading coefficients in its row echelon form that indicate linearly independent rows.
The equation ax=0 has only the trivial solution (x = 0), confirming the linear independence of columns.

Linear Independence and Transformation

The columns of A form a linearly independent set, meaning no column can be expressed as a linear combination of others.
The linear transformation defined as x -> Ax is one-to-one, implying distinct inputs map to distinct outputs.

Span and Basis

For every vector b in R^n, the equation ax = b has at least one solution, meaning the columns of A span the entirety of R^n.
The transformation x -> ax maps Rn onto Rn, signifying that every output is achievable from some input.

Basis, Column Space, and Dimensions

The columns of A not only span R^n but also form a basis, indicating that they are both spanning and linearly independent.
The column space of A, denoted as Col A, equals R^n, confirming that the set completely fills the n-dimensional space.

Dimensionality and Rank

The dimension of the column space is given by dim Col A = n, representing its maximal dimensionality.
The rank of the matrix A is n, indicating the maximum number of linearly independent column vectors it contains.

Null Space Properties

The null space of A, denoted as Nul A, contains only the zero vector, indicating no non-trivial solutions exist for the homogeneous system.
The dimension of the null space is dim Nul A = 0, reinforcing the idea that A is invertible as it lacks free variables.

Test your knowledge on essential concepts in Linear Algebra with these flashcards. Each card focuses on fundamental properties of invertible matrices and their characteristics. Perfect for students looking to reinforce their understanding of this key mathematical topic.