Difference between revisions of "Vectors and Matrices"
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| + | ==Matrices== | ||
| + | A matrix is an array of numbers arranged into rows and columns. Some examples of matrices are, | ||
| + | |||
| + | <math>A= | ||
| + | \begin{bmatrix} | ||
| + | 2 & 4 & 6 \\ 0 & -5 & 7.105\\ 1 & -3 & 2 | ||
| + | \end{bmatrix} | ||
| + | ,\quad | ||
| + | B= | ||
| + | \begin{bmatrix} | ||
| + | -3 & -4 \\ \pi & \sqrt{2} | ||
| + | \end{bmatrix} | ||
| + | ,\mbox{ and }\,\,\, | ||
| + | C= | ||
| + | \begin{bmatrix} | ||
| + | -3 & -5 & 0 & 0 \\ -1 & 4.56 & 3.28 & 19 | ||
| + | \end{bmatrix} | ||
| + | .</math> | ||
| + | |||
| + | When describing matrices we indicate the number of rows first, then the number of columns. For example, the matrix <math>C</math> with two rows and four columns is said to be a <math>2\times 4</math> matrix. | ||
| + | |||
| + | It is standard notation to name matrices with capital letters and to use lower case letters with subscripts to identify particular entries in a matrix. | ||
| + | |||
| + | For example, to identify the entry in row 1 and column 3 of matrix <math>A</math> we would write <math>a_{13}</math>. To indicate that this entry is a six we would write the equation <math>a_{13}=6</math>. | ||
| + | |||
| + | Two matrices are considered to be | ||
| + | '''equal''' only if they are the same size and every pair of corresponding elements are equal. | ||
| + | |||
| + | A '''column matrix''' is a matrix with only one column. Similarly, a '''row matrix''' has only one row. | ||
| + | ==Vectors== | ||
| + | A '''vector''' is an object often defined by a long list of properties. However, for now we will avoid the more complicated definition, and just say that a vector is an ordered list of numbers. Later we will see that vectors can really be much more. | ||
| + | |||
| + | An ordered pair, <math>(x, y)</math>, that is used to identify a point in the plane can be considered to be a vector. | ||
| + | |||
| + | Similarly, an ordered triple, <math>(x, y, z)</math> is a vector. | ||
| + | |||
| + | Obviously, row and column matrices can also be considered to be a vector. | ||
| + | |||
| + | It is common to name vectors using variables with arrows above. | ||
| + | |||
| + | For example, we might write | ||
| + | <math>\vec{v}=(2, 3, 5, -4),\mbox{ or } \vec{w}= | ||
| + | \begin{bmatrix} 4 \\ 5 \\ 0 | ||
| + | \end{bmatrix}</math>. | ||
| + | |||
| + | For the most part, it will convenient to think of vectors as column matrices. | ||
| + | |||
| + | |||
==Resources== | ==Resources== | ||
* [https://www.unf.edu/~mzhan/chapter3.pdf Brief Introduction to Vectors and Matrices], University of North Florida | * [https://www.unf.edu/~mzhan/chapter3.pdf Brief Introduction to Vectors and Matrices], University of North Florida | ||
* [https://www.usna.edu/Users/math/uhan/sm286a/lessons/02.pdf Introduction to Matrices and Vectors], United States Naval Academy | * [https://www.usna.edu/Users/math/uhan/sm286a/lessons/02.pdf Introduction to Matrices and Vectors], United States Naval Academy | ||
| + | |||
| + | ==Licensing== | ||
| + | Content obtained and/or adapted from: | ||
| + | * [https://en.wikibooks.org/wiki/Linear_Algebra/Matrices_and_Vectors/ Matrices and Vectors, Wikibooks: Linear Algebra] under a CC BY-SA license | ||
Latest revision as of 07:14, 3 November 2021
Contents
Matrices
A matrix is an array of numbers arranged into rows and columns. Some examples of matrices are,
When describing matrices we indicate the number of rows first, then the number of columns. For example, the matrix with two rows and four columns is said to be a matrix.
It is standard notation to name matrices with capital letters and to use lower case letters with subscripts to identify particular entries in a matrix.
For example, to identify the entry in row 1 and column 3 of matrix we would write . To indicate that this entry is a six we would write the equation .
Two matrices are considered to be equal only if they are the same size and every pair of corresponding elements are equal.
A column matrix is a matrix with only one column. Similarly, a row matrix has only one row.
Vectors
A vector is an object often defined by a long list of properties. However, for now we will avoid the more complicated definition, and just say that a vector is an ordered list of numbers. Later we will see that vectors can really be much more.
An ordered pair, , that is used to identify a point in the plane can be considered to be a vector.
Similarly, an ordered triple, is a vector.
Obviously, row and column matrices can also be considered to be a vector.
It is common to name vectors using variables with arrows above.
For example, we might write .
For the most part, it will convenient to think of vectors as column matrices.
Resources
- Brief Introduction to Vectors and Matrices, University of North Florida
- Introduction to Matrices and Vectors, United States Naval Academy
Licensing
Content obtained and/or adapted from:
- Matrices and Vectors, Wikibooks: Linear Algebra under a CC BY-SA license
