3 Linear Algebra Review

Matrices and Vectors

The dimension of the matrix is going to be written as the number of row times the number of columns in the matrix.

A vector turns out to be a special case of a matrix.

Matrix with just one column is what we call a vector.

Addition and Scalar Multiplication

Addition

\(
\begin{bmatrix}
1 & 0 \\
2 & 5 \\
3 & 1
\end{bmatrix}
+
\begin{bmatrix}
4 & 0.5 \\
2 & 5 \\
0 & 1
\end{bmatrix}
=
\begin{bmatrix}
5 & 0.5 \\
4 & 10 \\
3 & 2
\end{bmatrix}
\)

Scalar Multiplication

\(
3 *
\begin{bmatrix}
1 & 0 \\
2 & 5 \\
3 & 1
\end{bmatrix}
=
\begin{bmatrix}
3 & 0 \\
6 & 15 \\
9 & 3
\end{bmatrix}
=
\begin{bmatrix}
1 & 0 \\
2 & 5 \\
3 & 1
\end{bmatrix}
* 3
\)

Matrix Vector Multiplication

\(
\begin{bmatrix}
1 & 3 \\
4 & 0 \\
2 & 1
\end{bmatrix}
*
\begin{bmatrix}
1 \\
5
\end{bmatrix}
=
\begin{bmatrix}
16 = 1 * 1 + 3 * 5 \\
4 = 4 * 1 + 0 * 5 \\
7 = 2 * 1 + 1 * 5
\end{bmatrix}
\)

Matrix Matrix Multiplication

\(
\begin{bmatrix}
C0 & C1 \\
C2 & C3
\end{bmatrix}
=
\begin{bmatrix}
A0 & A1 \\
A2 & A3
\end{bmatrix}
*
\begin{bmatrix}
B0 & B1 \\
B2 & B3
\end{bmatrix}
\)

\(C0 = A0 * B0 + A1 * B2\)
\(C1 = A0 * B1 + A1 * B3\)
\(C2 = A2 * B0 + A3 * B2\)
\(C3 = A2 * B1 + A3 * B3\)

Matrix Multiplication Properties

\(A * B \neq B * A\)
\(A * (B * C) = (A * B) * C\)

Identity matrix, has the property that it has ones along the diagonals, right, and so on and is zero everywhere else.

\(AA^{-1} = A^{-1}A = I\)
\(AI = IA = A\)

Inverse and Transpose

Inverse

\(A^{-1}\)

Transpose

\(A^{T}\)

\(
\begin{vmatrix}
a & b\\
c & d\\
e & f
\end{vmatrix}
^{T}
=
\begin{vmatrix}
a & c & e\\
b & d & f
\end{vmatrix}
\)

Leave a Reply

Your email address will not be published. Required fields are marked *