hcju/proofwikips huggingface dataset:
Elementary Row Operation/Examples/lambda r2
== Examples of Elementary Row Operations ==  Consider the elementary row operation $e$ defined as: :$e := r_2 \to \lambda r_2$ acting on a matrix space $\map \MM {3, n}$ for some $n \in \Z_{>0}$. The elementary row matrix corresponding to $e$ is: :$\begin {pmatrix} 1 & 0 & 0 \\ 0 & \lambda & 0 \\ 0 & 0 & 1 \end {pmatrix}$  == Proof == Let $\mathbf E$ denote the elementary row matrix corresponding to $e$. From Elementary Matrix corresponding to Elementary Row Operation: Scale Row and Add: :$E_{a b} = \begin {cases} \delta_{a b} & : a \ne 2 \\ \lambda \cdot \delta_{a b} & : a = 2 \end{cases}$ where: :$E_{a b}$ denotes the element of $\mathbf E$ whose indices are $\tuple {a, b}$ :$\delta_{a b}$ is the Kronecker delta: ::$\delta_{a b} = \begin {cases} 1 & : \text {if $a = b$} \\ 0 & : \text {if $a \ne b$} \end {cases}$ That is: :When $a \ne 2$, the elements of $\mathbf E$ are $0$ except for those on the main diagonal, when they are $1$ :When $a = 2$, the elements of $\mathbf E$ are $0$ except for those on the main diagonal, when they are $\lambda$. Hence $\mathbf E$ can be constructed as described.  == Sources == * {{BookReference|Linear Algebra|1998|Richard Kaye|author2 = Robert Wilson|prev = Elementary Row Operation/Examples/r3 + 2r2|next = Elementary Row Operation/Examples/Swap r1 and r2}}: Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations: $2$
Elementary Row Operation/Examples/Swap r1 and r2
== Examples of Elementary Row Operations ==  Consider the elementary row operation $e$ defined as: :$e := r_1 \leftrightarrow r_2$ acting on a matrix space $\map \MM {3, n}$ for some $n \in \Z_{>0}$. The elementary row matrix corresponding to $e$ is: :$\begin {pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end {pmatrix}$  == Proof == By definition, the elementary row matrix corresponding to $e$ is found by applying $e$ to the unit matrix. By definition of unit matrix: :$\mathbf I = \begin {pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {pmatrix}$ Let $\mathbf E$ denote the elementary row matrix corresponding to $e$. $\mathbf E$ is constructed by exchanging row $1$ with row $2$.  == Sources == * {{BookReference|Linear Algebra|1998|Richard Kaye|author2 = Robert Wilson|prev = Elementary Row Operation/Examples/lambda r2|next = Definition:Row Operation}}: Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations: $3$
Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 1
Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 1 0 88443 474449 2020-06-17T06:35:30Z Prime.mover 59 Created page with "== Examples of Use of Matrix is Row Equivalent to Echelon Matrix ==  Let $\mathbf A = \begin {bmatrix} 0 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 0 \\ \end {bmatrix}$..." wikitext text/x-wiki == Examples of Use of Matrix is Row Equivalent to Echelon Matrix ==  Let $\mathbf A = \begin {bmatrix} 0 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 0 \\ \end {bmatrix}$ This can be converted into the echelon form: :$\mathbf E = \begin {bmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end {bmatrix}$  == Proof == Using Row Operation to Clear First Column of Matrix we obtain: :$\mathbf B = \begin {bmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \\ \end {bmatrix}$ which happens on the first step, exchanging row $1$ with row $3$. Then we investigate the submatrix: :$\mathbf B' = \begin {bmatrix} 1 & 0 \\ 1 & 1 \\ \end {bmatrix}$ Using Row Operation to Clear First Column of Matrix we obtain: :$\mathbf C' = \begin {bmatrix} 1 & 0 \\ 0 & 1 \\ \end {bmatrix}$ which is obtained by adding $-1$ of row $1$ of $\mathbf B'$ to row $2$ of $\mathbf B'$. This is the same as adding $-1$ of row $2$ of $\mathbf B$ to row $3$ of $\mathbf B$. Thus we are left with: :$\mathbf E = \begin {bmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end {bmatrix}$  == Sources == * {{BookReference|Linear Algebra|1998|Richard Kaye|author2 = Robert Wilson|prev = Matrix is Row Equivalent to Echelon Matrix/Examples|next = Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 2}}: Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations: Exercise $1.4 \ \text {(a)}$
Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 2
Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 2 0 88444 474450 2020-06-17T06:39:44Z Prime.mover 59 Created page with "== Examples of Use of Matrix is Row Equivalent to Echelon Matrix ==  Let $\mathbf A = \begin {bmatrix} 1 & 1 & 1 & 1 \\ 2 & 3 & 4 & 5 \\ 3 & 4 & 5 & 6 \\ \end..." wikitext text/x-wiki == Examples of Use of Matrix is Row Equivalent to Echelon Matrix ==  Let $\mathbf A = \begin {bmatrix} 1 & 1 & 1 & 1 \\ 2 & 3 & 4 & 5 \\ 3 & 4 & 5 & 6 \\ \end {bmatrix}$ This can be converted into the echelon form: :$\mathbf E = \begin {bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 \\ \end {bmatrix}$  == Proof == Using Row Operation to Clear First Column of Matrix we obtain: :$\mathbf B = \begin {bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 1 & 2 & 3 \\ \end {bmatrix}$ which is obtained by: :adding $-2$ of row $1$ to row $2$ :adding $-3$ of row $1$ to row $3$. Then we investigate the submatrix: :$\mathbf B' = \begin {bmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \\ \end {bmatrix}$ Using Row Operation to Clear First Column of Matrix we obtain: :$\mathbf C' = \begin {bmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ \end {bmatrix}$ which is obtained by adding $-1$ of row $1$ of $\mathbf B'$ to row $2$ of $\mathbf B'$. Thus we are left with: :$\mathbf E = \begin {bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 \\ \end {bmatrix}$  == Sources == * {{BookReference|Linear Algebra|1998|Richard Kaye|author2 = Robert Wilson|prev = Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 1|next = Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 3}}: Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations: Exercise $1.4 \ \text {(b)}$
Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 3
Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 3 0 88445 474451 2020-06-17T06:52:35Z Prime.mover 59 Created page with "== Examples of Use of Matrix is Row Equivalent to Echelon Matrix ==  Let $\mathbf A = \begin {bmatrix} 1 & 2 & 3 & 5 \\ 1 & 2 & 3 & 4 \\ 0 & 0 & 1 & 1 \\ \end..." wikitext text/x-wiki == Examples of Use of Matrix is Row Equivalent to Echelon Matrix ==  Let $\mathbf A = \begin {bmatrix} 1 & 2 & 3 & 5 \\ 1 & 2 & 3 & 4 \\ 0 & 0 & 1 & 1 \\ \end {bmatrix}$ This can be converted into the echelon form: :$\mathbf E = \begin {bmatrix} 1 & 2 & 3 & 5 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \\ \end {bmatrix}$  == Proof == Using Row Operation to Clear First Column of Matrix we obtain: :$\mathbf B = \begin {bmatrix} 1 & 2 & 3 & 5 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 1 \\ \end {bmatrix}$ which is obtained by adding $-2$ of row $1$ to row $2$. Then we investigate the submatrix: :$\mathbf B' = \begin {bmatrix} 0 & 0 & -1 \\ 0 & 1 & 1 \\ \end {bmatrix}$ Using Row Operation to Clear First Column of Matrix we obtain: :$\mathbf C' = \begin {bmatrix} 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end {bmatrix}$ which is obtained by: :$(1): \quad$ exchanging row $1$ of $\mathbf B'$ with row $2$ of $\mathbf B'$. :$(2): \quad$ multiplying row $2$ of $\mathbf B'$ by $-1$. Thus we are left with: :$\mathbf E = \begin {bmatrix} 1 & 2 & 3 & 5 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \\ \end {bmatrix}$  == Also presented as == Some sources use the non-unity variant of the echelon matrix. Such sources do not require that the leading coefficients necessarily have to equal to $1$. Hence they consider the final step to convert row $3$ of $\mathbf E$ from $\begin {bmatrix} 0 & 0 & 0 & -1 \end {bmatrix}$ to $\begin {bmatrix} 0 & 0 & 0 & 1 \end {bmatrix}$ as optional. == Sources == * {{BookReference|Linear Algebra|1998|Richard Kaye|author2 = Robert Wilson|prev = Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 2|next = Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 4}}: Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations: Exercise $1.4 \ \text {(c)}$
Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 4
Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 4 0 88446 474452 2020-06-17T06:57:05Z Prime.mover 59 Created page with "== Examples of Use of Matrix is Row Equivalent to Echelon Matrix ==  Let $\mathbf A = \begin {bmatrix} 1 & 1 & 2 & 3 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 3 & 3 \\ \end..." wikitext text/x-wiki == Examples of Use of Matrix is Row Equivalent to Echelon Matrix ==  Let $\mathbf A = \begin {bmatrix} 1 & 1 & 2 & 3 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 3 & 3 \\ \end {bmatrix}$ This can be converted into the echelon form: :$\mathbf E = \begin {bmatrix} 1 & 1 & 2 & 3 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ \end {bmatrix}$  == Proof == It is noted that $\mathbf A$ is already most of the way there. It remains to use the elementary row operation: :$e := r_3 \to r_3 - 3 r_2$ to convert $\mathbf A$ to the form: :$\mathbf E = \begin {bmatrix} 1 & 1 & 2 & 3 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ \end {bmatrix}$  == Sources == * {{BookReference|Linear Algebra|1998|Richard Kaye|author2 = Robert Wilson|prev = Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 3|next = Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 5}}: Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations: Exercise $1.4 \ \text {(d)}$
Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 5
== Examples of Use of Matrix is Row Equivalent to Echelon Matrix ==  Let $\mathbf A = \begin {bmatrix} -1 & 0 & 1 & 2 & 3 \\ 0 & 1 & 2 & 3 & 4 \\ -1 & -2 & -3 & -4 & -5 \\ \end {bmatrix}$ This can be converted into the echelon form: :$\mathbf E = \begin {bmatrix} 1 & 0 & -1 & -2 & -3 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 0 & 0 & 0 \\ \end {bmatrix}$  == Proof == This matrix can more easily be handled by direct application of elementary row operations, as follows. Let $e_1$ be the elementary row operation: :$e_1 := r_3 \to r_3 - r_1$ which leaves: $\mathbf A_1 = \begin {bmatrix} -1 & 0 & 1 & 2 & 3 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & -2 & -4 & -6 & -8 \\ \end {bmatrix}$ Let $e_2$ be the elementary row operation: :$e_2 := r_3 \to r_3 + 2 r_2$ which leaves: :$\mathbf A_2 = \begin {bmatrix} -1 & 0 & 1 & 2 & 3 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 0 & 0 & 0 \\ \end {bmatrix}$ $\mathbf A_2$ is in non-unity echelon form. It remains to perform the elementary row operation $e_3$ to convert it into echelon form: :$e_3 := r_1 \to -r_1$ which leaves: :$\mathbf E = \begin {bmatrix} 1 & 0 & -1 & -2 & -3 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 0 & 0 & 0 \\ \end {bmatrix}$  == Also presented as == Some sources use the non-unity variant of the echelon matrix. Such sources do not require that the leading coefficients necessarily have to equal to $1$. Hence they consider the final step to convert row $1$ of $\mathbf E$ from $\begin {bmatrix} -1 & 0 & 1 & 2 & 3 \end {bmatrix}$ to $\begin {bmatrix} 1 & 0 & -1 & -2 & -3 \end {bmatrix}$ as optional. == Sources == * {{BookReference|Linear Algebra|1998|Richard Kaye|author2 = Robert Wilson|prev = Matrix is Row Equivalent to Echelon Matrix/Examples/Arbitrary Matrix 4|next = Definition:Rank/Matrix/Definition 2}}: Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations: Exercise $1.4 \ \text {(e)}$
Simultaneous Linear Equations/Examples/Arbitrary System 1
== Example of Simultaneous Linear Equations ==  The system of simultaneous linear equations: {{begin-eqn}} {{eqn | n = 1       | l = x_1 - 2 x_2 + x_3       | r = 1 }} {{eqn | n = 2       | l = 2 x_1 - x_2 + x_3       | r = 2 }} {{eqn | n = 3       | l = 4 x_1 + x_2 - x_3       | r = 1 }} {{end-eqn}} has as its solution set: {{begin-eqn}} {{eqn | l = x_1       | r = -\dfrac 1 2 }} {{eqn | l = x_2       | r = \dfrac 1 2 }} {{eqn | l = x_3       | r = \dfrac 3 2 }} {{end-eqn}}  == Proof == Subtract $2 \times$ equation $(1)$ from equation $(2)$. Subtract $4 \times$ equation $(1)$ from equation $(3)$. This gives us: {{begin-eqn}} {{eqn | n = 1       | l = x_1 - 2 x_2 + x_3       | r = 1 }} {{eqn | n = 2'       | l = 3 x_2 - x_3       | r = 0 }} {{eqn | n = 3'       | l = 9 x_2 - 5 x_3       | r = -3 }} {{end-eqn}} Divide equation $(2')$ by $3$ to get $(2'')$. Add $2 \times$ equation $(2'')$ to equation $(1)$. Subtract $9 \times$  equation $(2'')$ from equation $(3')$. This gives us: {{begin-eqn}} {{eqn | n = 1'       | l = x_1 + \dfrac {x_3} 3       | r = 1 }} {{eqn | n = 2''       | l = x_2 - \dfrac {x_3} 3       | r = 0 }} {{eqn | n = 3''       | l = -2 x_3       | r = -3 }} {{end-eqn}} From equation $(3'')$ we have directly that $x_3 = \dfrac 3 2$. Substituting for $x_3$ in equation $(1')$ and equation $(2'')$ gives the single solution: {{begin-eqn}} {{eqn | l = x_1       | r = -\dfrac 1 2 }} {{eqn | l = x_2       | r = \dfrac 1 2 }} {{eqn | l = x_3       | r = \dfrac 3 2 }} {{end-eqn}}  == Sources == * {{BookReference|Linear Algebra: An Introduction|1982|A.O. Morris|ed = 2nd|edpage = Second Edition|next = Simultaneous Linear Equations/Examples/Arbitrary System 2}}: Chapter $1$: Linear Equations and Matrices: $1.1$ Introduction: Example $\text {(i)}$
Simultaneous Linear Equations/Examples/Arbitrary System 2
== Example of Simultaneous Linear Equations ==  The system of simultaneous linear equations: {{begin-eqn}} {{eqn | n = 1       | l = x_1 + x_2       | r = 2 }} {{eqn | n = 2       | l = 2 x_1 + 2 x_2       | r = 3 }} {{end-eqn}} has no solutions.  == Proof ==  $(1)$ and $(2)$ together have a solution. Subtract $2 \times$ equation $(1)$ from equation $(2)$. {{begin-eqn}} {{eqn | n = 1       | l = x_1 - 2 x_2 + x_3       | r = 1 }} {{eqn | n = 2'       | l = 0       | r = -1 }} {{end-eqn}} which is an inconsistency. Hence there is no such solution.  == Sources == * {{BookReference|Linear Algebra: An Introduction|1982|A.O. Morris|ed = 2nd|edpage = Second Edition|prev = Simultaneous Linear Equations/Examples/Arbitrary System 1|next = Simultaneous Linear Equations/Examples/Arbitrary System 3}}: Chapter $1$: Linear Equations and Matrices: $1.1$ Introduction: Example $\text {(ii)}$
Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 1
== Corollary to Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations ==  Let $S$ be a system of simultaneous linear equations: :$\displaystyle \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$ Let $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ denote the augmented matrix of $S$. Let $\begin {pmatrix} \mathbf R & \mathbf s \end {pmatrix}$ be a reduced echelon matrix derived from $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$. Let $S'$ be the system of simultaneous linear equations: :$\displaystyle \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \rho_{i j} x_j = \sigma_i$ whose augmented matrix is $\begin {pmatrix} \mathbf R & \mathbf s \end {pmatrix}$. Then $S$ and $S'$ are equivalent.  == Proof == By Matrix is Row Equivalent to Reduced Echelon Matrix, $\begin {pmatrix} \mathbf R & \mathbf s \end {pmatrix}$ can be obtained from $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ by means of a finite sequence of elementary row operations. By Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations, the system of simultaneous linear equations whose augmented matrix is obtained from $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ in this way is equivalent to $S$ after each elementary row operation. Hence the entire sequence of such systems of simultaneous linear equations are equivalent to $S$. In particular, this applies to $S'$.  == Sources == * {{BookReference|Linear Algebra: An Introduction|1982|A.O. Morris|ed = 2nd|edpage = Second Edition|prev = Definition:Inverse of Elementary Row Operation|next = Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 2}}: Chapter $1$: Linear Equations and Matrices: $1.3$ Applications to Linear Equations: Corollary $1$
Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 3
== Corollary to Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations ==  Let $S$ be a system of homogeneous simultaneous linear equations: :$\displaystyle \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = 0$ If $m < n$, then $S$ has at least one non-trivial solution.  == Proof == Let $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ denote the augmented matrix of $S$. Because $S$ is homogeneous, we have that $\mathbf b = \mathbf 0$, and so its augmented matrix is $\begin {pmatrix} \mathbf A & \mathbf 0 \end {pmatrix}$. Let $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$ be a reduced echelon matrix derived from $\begin {pmatrix} \mathbf A & \mathbf 0 \end {pmatrix}$. Let $S'$ be the system of simultaneous linear equations: :$\displaystyle \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \rho_{i j} x_j = 0$ whose augmented matrix is $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$. By Corollary 1, $S$ and $S'$ are equivalent. Hence any every solution to $S'$ is also a solution to $S$. Consider the structure of $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$. Suppose the leading coefficients appear in columns which we name $j_1, j_2, \ldots, j_l$. Let the remaining columns be named $j_{l + 1}, j_{l + 2}, \ldots, j_n$. Then we have that $S'$ can be expressed as: {{begin-eqn}} {{eqn | l = x_{j_1} + \sum_{k \mathop = l + 1}^n \rho_{i j_k} x_{j_k}       | r = 0 }} {{eqn | l = x_{j_2} + \sum_{k \mathop = l + 1}^n \rho_{2 j_k} x_{j_k}       | r = 0 }} {{eqn | o = \cdots }} {{eqn | l = x_{j_l} + \sum_{k \mathop = l + 1}^n \rho_{2 j_k} x_{j_k}       | r = 0 }} {{end-eqn}} where $l + 1 \le m$. Setting arbitrary values to $x_{j_k}$ for $l < k \le n$ gives us a non-trivial solution for $S$.  == Sources == * {{BookReference|Linear Algebra: An Introduction|1982|A.O. Morris|ed = 2nd|edpage = Second Edition|prev = Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 2|next = Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 4}}: Chapter $1$: Linear Equations and Matrices: $1.3$ Applications to Linear Equations: Corollary $3$
Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 4
Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 4 0 88539 474887 2020-06-20T12:54:48Z Prime.mover 59 Created page with "== Corollary to Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations ==  Let $S$ be a system of Definiti..." wikitext text/x-wiki == Corollary to Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations ==  Let $S$ be a system of homogeneous simultaneous linear equations: :$\displaystyle \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = 0$ Let $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$ be a reduced echelon matrix derived from $\begin {pmatrix} \mathbf A & \mathbf 0 \end {pmatrix}$. Let the number of non-zero rows of $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$ be $l$. If $l = n$, then the only solution to $S$ is the trivial solution.  == Proof == Consider the structure of $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$. Suppose the leading coefficients appear in columns which we name $j_1, j_2, \ldots, j_n$. As there are $n$ columns as well as $n$ non-zero rows: :each row has exactly one $1$ in it :each column has exactly one $1$ in it. Thus $S'$ can be expressed as: {{begin-eqn}} {{eqn | l = x_{j_1}       | r = 0 }} {{eqn | l = x_{j_2}       | r = 0 }} {{eqn | o = \cdots }} {{eqn | l = x_{j_n}       | r = 0 }} {{end-eqn}} and the result follows.  == Sources == * {{BookReference|Linear Algebra: An Introduction|1982|A.O. Morris|ed = 2nd|edpage = Second Edition|prev = Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 3|next = Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 5}}: Chapter $1$: Linear Equations and Matrices: $1.3$ Applications to Linear Equations: Corollary $3$
Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse
Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse 0 88767 476442 476440 2020-06-30T09:23:35Z Prime.mover 59 Replaced content with "== Theorem ==  Let $e$ be an elementary row operation.  Let $\mathbf E$ be the Definition:Elementary Row Matrix|elem..." wikitext text/x-wiki == Theorem ==  Let $e$ be an elementary row operation. Let $\mathbf E$ be the elementary row matrix corresponding to $e$. Let $e'$ be the inverse of $e$. Then the elementary row matrix corresponding to $e'$ is the inverse of $\mathbf E$.  == Proof 1 == {{:Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse/Proof 1}} == Proof 2 == {{:Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse/Proof 2}} == Also see == * Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse
Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse
Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse 0 88788 476460 2020-06-30T10:39:05Z Prime.mover 59 Created page with "== Theorem ==  Let $e$ be an elementary column operation.  Let $\mathbf E$ be the Definition:Elementary Column Matrix..." wikitext text/x-wiki == Theorem ==  Let $e$ be an elementary column operation. Let $\mathbf E$ be the elementary column matrix corresponding to $e$. Let $e'$ be the inverse of $e$. Then the elementary column matrix corresponding to $e'$ is the inverse of $\mathbf E$.  == Proof 1 == {{:Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse/Proof 1}} == Proof 2 == {{:Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse/Proof 2}} == Also see == * Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse
Elementary Column Operations as Matrix Multiplications/Corollary
Elementary Column Operations as Matrix Multiplications/Corollary 0 88796 476469 2020-06-30T13:33:38Z Prime.mover 59 Created page with "== Theorem ==  Let $\mathbf X$ and $\mathbf Y$ be two $m \times n$ matrices that differ by exactly one Definition:Elementary Column Operat..." wikitext text/x-wiki == Theorem ==  Let $\mathbf X$ and $\mathbf Y$ be two $m \times n$ matrices that differ by exactly one elementary column operation. Then there exists an elementary column matrix $\mathbf E$ of order $n$ such that: :$\mathbf X \mathbf E = \mathbf Y$  == Proof == Let $e$ be the elementary column operation such that $e \paren {\mathbf X} = \mathbf Y$. Then this result follows immediately from Elementary Column Operations as Matrix Multiplications: :$e \paren {\mathbf X} = \mathbf X \mathbf E = \mathbf Y$ where $\mathbf E = e \paren {\mathbf I}$.