Linear algebra is a branch of mathematics that studies vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. It provides a framework for understanding how to solve linear equations and perform vector operations, which are fundamental in various scientific and engineering disciplines.<\/p>\n\n\n\n
Vector Spaces<\/p>\n\n\n\n
A vector space \\( V \\) over a field \\( \\mathbb{{F}} \\) (such as the real numbers \\( \\mathbb{{R}} \\) or complex numbers \\( \\mathbb{{C}} \\)) is a set equipped with two operations: vector addition and scalar multiplication. These operations must satisfy the following axioms:<\/p>\n\n\n\n
1. Associativity of addition: For all \\( \\mathbf{{u}}, \\mathbf{{v}}, \\mathbf{{w}} \\in V \\),<\/p>\n\n\n\n
$$ (\\mathbf{{u}} + \\mathbf{{v}}) + \\mathbf{{w}} = \\mathbf{{u}} + (\\mathbf{{v}} + \\mathbf{{w}}) $$<\/p>\n\n\n\n
2. Commutativity of addition: For all \\( \\mathbf{{u}}, \\mathbf{{v}} \\in V \\),<\/p>\n\n\n\n
$$ \\mathbf{{u}} + \\mathbf{{v}} = \\mathbf{{v}} + \\mathbf{{u}} $$<\/p>\n\n\n\n
3. Identity element of addition: There exists an element \\( \\mathbf{{0}} \\in V \\) such that for all \\( \\mathbf{{u}} \\in V \\),<\/p>\n\n\n\n
$$ \\mathbf{{u}} + \\mathbf{{0}} = \\mathbf{{u}} $$<\/p>\n\n\n\n
4. Inverse elements of addition: For each \\( \\mathbf{{u}} \\in V \\), there exists an element \\( -\\mathbf{{u}} \\in V \\) such that<\/p>\n\n\n\n
$$ \\mathbf{{u}} + (-\\mathbf{{u}}) = \\mathbf{{0}} $$<\/p>\n\n\n\n
5. Compatibility of scalar multiplication with field multiplication: For all \\( a, b \\in \\mathbb{{F}} \\) and \\( \\mathbf{{u}} \\in V \\),<\/p>\n\n\n\n
$$ a(b\\mathbf{{u}}) = (ab)\\mathbf{{u}} $$<\/p>\n\n\n\n
6. Identity element of scalar multiplication: For all \\( \\mathbf{{u}} \\in V \\),<\/p>\n\n\n\n
$$ 1\\mathbf{{u}} = \\mathbf{{u}} $$<\/p>\n\n\n\n
7. Distributivity of scalar multiplication with respect to vector addition: For all \\( a \\in \\mathbb{{F}} \\) and \\( \\mathbf{{u}}, \\mathbf{{v}} \\in V \\),<\/p>\n\n\n\n
$$ a(\\mathbf{{u}} + \\mathbf{{v}}) = a\\mathbf{{u}} + a\\mathbf{{v}} $$<\/p>\n\n\n\n
8. Distributivity of scalar multiplication with respect to field addition: For all \\( a, b \\in \\mathbb{{F}} \\) and \\( \\mathbf{{u}} \\in V \\),<\/p>\n\n\n\n
$$ (a + b)\\mathbf{{u}} = a\\mathbf{{u}} + b\\mathbf{{u}} $$<\/p>\n\n\n\n
Linear Transformations<\/p>\n\n\n\n
A linear transformation (or linear map) is a function \\( T: V \\to W \\) between two vector spaces \\( V \\) and \\( W \\) that preserves the operations of vector addition and scalar multiplication. This means that for all \\( \\mathbf{{u}}, \\mathbf{{v}} \\in V \\) and all scalars \\( c \\in \\mathbb{{F}} \\):<\/p>\n\n\n\n
$$ T(\\mathbf{{u}} + \\mathbf{{v}}) = T(\\mathbf{{u}}) + T(\\mathbf{{v}}) $$<\/p>\n\n\n\n
$$ T(c\\mathbf{{u}}) = cT(\\mathbf{{u}}) $$<\/p>\n\n\n\n
Matrices<\/p>\n\n\n\n
Matrices are a convenient way to represent linear transformations. An \\( m \\times n \\) matrix is a rectangular array of numbers with \\( m \\) rows and \\( n \\) columns. The entry in the \\( i \\)-th row and \\( j \\)-th column of a matrix \\( A \\) is denoted by \\( a_{{ij}} \\). If \\( A \\) is an \\( m \\times n \\) matrix, and \\( \\mathbf{{x}} \\) is a vector in \\( \\mathbb{{F}}^n \\), the product \\( A\\mathbf{{x}} \\) is a vector in \\( \\mathbb{{F}}^m \\) defined by:<\/p>\n\n\n\n
$$ (A\\mathbf{{x}})_i = \\sum_{{j=1}}^n a_{{ij}} x_j $$<\/p>\n\n\n\n
for \\( i = 1, 2, \\ldots, m \\).<\/p>\n\n\n\n
Determinants<\/p>\n\n\n\n
The determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the matrix. For a \\( 2 \\times 2 \\) matrix \\( A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\), the determinant is defined as:<\/p>\n\n\n\n
$$ \\det(A) = ad – bc $$<\/p>\n\n\n\n
For a \\( 3 \\times 3 \\) matrix \\( A = \\begin{pmatrix} a & b & c \\\\ d & e & f \\\\ g & h & i \\end{pmatrix} \\), the determinant is defined as:<\/p>\n\n\n\n
$$ \\det(A) = aei + bfg + cdh – ceg – bdi – afh $$<\/p>\n\n\n\n
Eigenvalues and Eigenvectors<\/p>\n\n\n\n
An eigenvector of a square matrix \\( A \\) is a nonzero vector \\( \\mathbf{{v}} \\) such that multiplication by \\( A \\) alters only the scale of \\( \\mathbf{{v}} \\):<\/p>\n\n\n\n
$$ A\\mathbf{{v}} = \\lambda\\mathbf{{v}} $$<\/p>\n\n\n\n
where \\( \\lambda \\) is a scalar known as the eigenvalue corresponding to the eigenvector \\( \\mathbf{{v}} \\). To find the eigenvalues of a matrix \\( A \\), we solve the characteristic equation:<\/p>\n\n\n\n
$$ \\det(A – \\lambda I) = 0 $$<\/p>\n\n\n\n
where \\( I \\) is the identity matrix of the same dimension as \\( A \\).<\/p>\n","protected":false},"excerpt":{"rendered":"
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