8 Applications of Linear Algebra Free Online FlashCards

What does a vector represent?

A vector is an ordered list of numbers representing quantities, coordinates, states, or unknowns.

What is a matrix used for?

A matrix organizes coefficients or represents a rule that transforms input vectors into output vectors.

How is each entry of $A\mathbf{x}$ computed?

In a product model, the entry in row $i$ of $A\mathbf{x}$ is the dot product of row $i$ of $A$ with $\mathbf{x}$.

What is the purpose of Gaussian elimination?

Gaussian elimination transforms the augmented matrix $[A\mid\mathbf{b}]$ using elementary row operations to reveal pivot and free variables.

When does a linear system have infinitely many solutions?

A consistent system has infinitely many solutions when at least one free variable remains.

What determinant condition guarantees invertibility?

A square matrix is invertible precisely when $\det(A)\neq 0$.

What is the determinant formula for a $2\times2$ matrix?

For $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$, $\det(A)=ad-bc$.

What geometric information does a determinant provide?

For a two-dimensional transformation, $|\det(A)|$ is the area-scaling factor; a negative determinant also reverses orientation.

What conditions define a linear transformation?

A linear transformation preserves vector addition and scalar multiplication: $T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})$ and $T(c\mathbf{u})=cT(\mathbf{u})$.

How is composition of linear transformations represented?

If $T_1(\mathbf{x})=A_1\mathbf{x}$ and $T_2(\mathbf{x})=A_2\mathbf{x}$, then $(T_2\circ T_1)(\mathbf{x})=A_2A_1\mathbf{x}$.

What is the kernel of a linear transformation?

The kernel is the set of inputs mapped to zero: $\ker(T)=\{\mathbf{x}:T(\mathbf{x})=\mathbf{0}\}$.

What is a basis, and what does its size determine?

A basis is a linearly independent spanning set, and the number of vectors in it is the space's dimension.