Linear Algebra

 Linear Algebra Solver 

Linear Algebra

101. Matrix Addition: $C=A+B$

102. Matrix Multiplication: $C=AB$

103. Determinant of a 2x2 Matrix: $\det (A)=ad-bc$

104. Determinant of a 3x3 Matrix: $\det (A)=a(ei-fh)-b(di-fg)+c(dh-eg)$

105. Inverse of a 2x2 Matrix: $A^{-1}=\frac{1}{\det (A)}\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)$

106. Eigenvalues: $\det (A-\lambda I)=0$

107. Eigenvectors: $(A-\lambda I)\mathbf{v}=0$

108. Dot Product: $\mathbf{a}\cdot \mathbf{b}=a_1{b}_1+a_2{b}_2+\cdots +a_n{b}_n$

109. Cross Product: $\mathbf{a}\times \mathbf{b}=\mid \begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\mid$

110. Vector Magnitude: $\mathrm{\parallel }\mathbf{a}\mathrm{\parallel }=\sqrt{a_1^2+a_2^2+\cdots +a_n^2}$

111. Vector Projection: ${\text{proj}}_{\mathbf{b}}\mathbf{a}=\frac{\mathbf{a}\cdot \mathbf{b}}{\mathrm{\parallel }\mathbf{b}{\mathrm{\parallel }}^2}\mathbf{b}$

112. Matrix Transpose: $A^T$

113. Trace of a Matrix: $\text{tr}(A)=\sum a_{ii}$

114. Rank of a Matrix: Number of linearly independent rows/columns.

115. Null Space: $\text{Null}(A)=\{\mathbf{x}\mathrm{\mid }A\mathbf{x}=0\}$

116. Column Space: $\text{Col}(A)=\text{Span}(\text{columns of }A)$

117. Row Space: $\text{Row}(A)=\text{Span}(\text{rows of }A)$

118. Orthogonal Vectors: $\mathbf{a}\cdot \mathbf{b}=0$

119. Orthogonal Matrix: $A^{T}A=I$

120. Singular Value Decomposition: $A=U\mathrm{\Sigma }{V}^T$