Determine whether the set of vectors in M2,2 is linearly...

Determine whether the set of vectors in $M_{2,2}$ is linearly independent or linearly dependent.
$A=\left[\begin{array}{rr}1 & -1 \\ 4 & 5\end{array}\right], B=\left[\begin{array}{rr}4 & 3 \\ -2 & 3\end{array}\right], C=\left[\begin{array}{rr}1 & -8 \\ 22 & 23\end{array}\right]$

Key Concepts

Vector Spaces

Vector spaces are algebraic structures consisting of vectors, which can be added together and multiplied by scalars while satisfying certain axioms such as associativity, commutativity, and the existence of an additive identity and inverses. This concept applies to various types of objects including matrices.

Systems of Linear Equations

Determining whether a set of vectors is linearly independent often requires setting up and solving a system of linear equations that arise from the condition that a linear combination of the vectors equals the zero vector. If the system has only the trivial solution, the vectors are independent; otherwise, they are dependent.

Linear Combination

A linear combination involves taking several vectors and multiplying each by a scalar, then summing the results. The idea is fundamental in determining linear dependence or independence because if a set of vectors can be combined in a nontrivial way (with at least one scalar nonzero) to yield the zero vector, then the set is linearly dependent.

Linear Independence

Linear independence is a property of a set of vectors in which no vector can be written as a linear combination of the others. This means that the only solution to the equation formed by their linear combination equaling the zero vector is the trivial solution, where all coefficients are zero.

Matrices as Vectors

Matrices can be treated as elements of a vector space by considering them as vectors, typically by arranging their entries into a single column or row. For example, a 2x2 matrix can be represented as a vector in a 4-dimensional space, which allows for the application of linear algebra techniques such as examining linear dependence.

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